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Scalar

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis

Let be a cyclic group, is the element in the group, is the scalar, and the Multi-Scalar Multiplication (MSM) is the algorithm for calculating , the sum of multiple scalar multiplications. As the group operation is more complicated than the addition and multiplication of elements in the finite field, the MSM algorithm is employed to reduce the number of times of group operation as much as possible. Usually, is a cyclic group defined on the elliptic curve of on the finite field of , with the order being bits, and . If the group operation is conducted by the plain algorithm of fast exponentiation, the number of group operations needed for every is times on average, and for all is times in total. Next, we will introduce some optimization methods corresponding to larger . G P_i ∈ G a_i ∈ Z ∑_{i=1}^{n} a_{i} P_{i} G y^2=x^3+ax+b F_p | | G b ≈ 256 0 ≤ a_i< 2^b a_i.P_i 1.5bn a_i.P_i 1.5b n Optimization Based on the Windowing Technique We can split the bit scalar into c width windows: reduce the scalar to base system. b 2^c Here, } , so we reduce the original algorithm issue of -bit MSM into a smaller -bit issue. We can put the same scalars from into a bucket and write them in another form: Q_j=∑_{i=1}^{n a_{i j} P_{i} b c ∑_{i=1}^{n} a_{i j} P_{i} For example, when , it is calculated as: c = 3 When calculating , set the partial sum T_l=S_l+⋯+S_k, then ∑kS_k However, the operation results of can be obtained through the recurrence sequence . It means that only one group operation is needed for the operation for each . Under the MSM algorithm with cc-bit scalar, all requires times of group operation. All requires times and the sum of these requires times of group operation. Therefore, each window with the value of c requires times of group operations. The operation of } only requires the normal method of fast exponentiation by times of group operation. T_l T_l=T_l+1+S_l T_l S_k n−2^c+1 T_k 2^ 1 c− T_k ^c−2 2 n+2^c−2 ∑_{j=0}^{b / c-1} Q_{j} 2^{j c (b/c−1)(c+1)=b − c + b/c−1 In summary, we can conclude that the total number of group operations required by the method of the Windowing Technique of a width of is c Set , then the number of group operations is . When , the number of group operations reduces to of its original value. c = logn 2bn/logn n≈10^5 1/12 Optimization Based on Group Endomorphism For the cyclic group on the elliptic curve on the finite field F_p, if the following group endomorphism of φ can be found: there exists , satisfying holds for all the points on , then it is easy to prove that such an endomorphism is a multiplication map, namely there is a making hold for all the points of on , which means that whenever we know the coordinates of one point, we can change it to the coordinates of another point simply by multiplying a number in to both the values of abscissa and ordinate. The algorithm can be further optimized based on this important property. G y^2=x^3+ax+b α,β ∈ F_p φ(x,y)=(αx,βy) G λ φ(P)=λP P G F_p When . The reverse value of one point can be obtained only by taking the opposite number from the ordinate. Based on this feature, in the plain fast exponentiation algorithm, the scalar can be written into the non-adjacent form (NAF), namely and any two adjacent cannot be non-zero at the same time. Compared to the -bit scalars, the group operation number in the fast exponentiation algorithm can be reduced from the original average of times to times. This technique can also be used to optimize the Windowing Technique. After writing to λ = −1, α = 1, β = −1 e_{i}.2^{i}, e_{i} ∈ {−1,0,1} ∑ e_i b 3/2⋅b 4/3⋅b a_i a_i=∑_{j=0}^{b / c-1} a_{i, j} 2^{j c}, 0 <= a_{i, j}<2^{c} conduct the following operation: The result of and - can be obtained. Since the complexity of addition and subtraction in group operation of an elliptic curve is the same, we can put these terms into a bucket according to the absolute value of scalars. Therefore, the number of groups can be reduced from the original 2c to , and the overall number of group operations required is reduced from (1) to: a_i=∑_{j=0}^{b / c-1} a’_{i, j}^2^{j c} 2^{c-1} <= a’_{i, j}^ <= 2^{c-1} 2^c If the parameters of the elliptic curve are special, for example, the BLS curve can be written as , and , take third-order element α from F_p^x, then there is a corresponding that holds y^2=x^3+b p ≡ 1( mod3 ) λ λ( )=( ) x,y αx,y Algorithm 1: Signed Digits In addition, . Then the multiplication operation can be optimized as follows: λ3 ≡ 1(mod |G| ) From [1], we can find a small enough scalar | |, | |<√ to make the above equation true, which gives us a way to reduce the multiplication of bb bits into the multiplication of two bits. Use it into the Windowing Technique, and the number of group operations is reduced to m1 m2 G b/2 In summary, both the above two optimization methods can reduce the number of group operations by when . 5.5% n=10^5 References [1] Francesco Sica, Mathieu Ciet, and Jean-Jacques Quisquater. Analysis of the gallant-lambert-vanstone method based on eﬀicient endomorphisms: Elliptic and hyperelliptic curves. In International Workshop on Selected Areas in Cryptography, pages 21–36. Springer, 2002.

Optimization of Multi-Scalar Multiplication Algorithm: Sin7Y Tech Review (21)

Authors:
(1) Sergey Kucheryavskiy, Department of Chemistry and Bioscience, Aalborg University and a Corresponding author (svk@bio.aau.dk);
(2) Sergei Zhilin, CSort, LLC., Germana Titova st. 7, Barnaul, 656023, Russia and Contributing authors0 (szhilin@gmail.com). Editor's note: This is Part 2 of 4 of a study detailing a new method for the augmentation of numeric and mixed datasets. Read the rest below. Table of Links Abstract and 1 Introduction
2 Methods

2.1 Generation of PV-sets based on Singular Value Decomposition
2.2 Generation of PV-sets based on PLS decomposition


3 Results

3.1 Datasets
3.2 ANN regression of Tecator data
3.3 ANN classification of Heart data


4 Discussion

5 Conclusions and References 2 Methods This chapter presents the general idea behind the Procrustes cross-validation approach, provides the necessary mathematical background and describes two implementations of the approach in detail. Let {X, Y} be two matrices of size I × J and I × M correspondingly, representing the original training set to be augmented. Columns of matrix X are predictors — variables measured or observed directly, for example, light absorbance at different wavelengths in the case of spectra, abundance of operational taxonomic units (OTU) in the case of 16S RNA sequencing, results of clinical tests for patients etc. If the original dataset has qualitative predictors, they should be replaced by corresponding dummy variables, which are combined with quantitative predictors to form X. Columns of matrix Y are responses — variables whose values should be predicted. There are situations when matrix Y is obsolete, for example, in the case of developing of one class classifier, only matrix with predictors is needed. To augment the data a specific model should be developed. It aims at capturing the variance-covariance structure of X, or its part, directly related to the variance of values in Y. This model will be referred to as the PV-model and denoted as M. The selection of the methods for creating the PV-model depends on the objectives. Thus, if data augmentation is needed for developing a one class classifier, when the matrix with responses, Y, is obsolete, we suggest using singular value decomposition (SVD), which is efficient in capturing the whole variance-covariance structure of X. In the case of augmenting data for discrimination models, there can be two solutions. One solution is to consider each class as an independent population and use SVD decomposition to capture the variance-covariance structure of X within each class. The second solution will be to create a matrix with dummy predictors, Y, where each column corresponds to a particular class, and employ PLS decomposition. More details about using SVD and PLS decompositions are provided in the next two sections. A general definition of the approach can be formulated as follows. Let us create a global PV-model M by fitting all data points from the training set {X, Y}. If this model is then applied to a new dataset, {X∗ , Y∗}, it will result in a set of outcomes R∗: M(X∗ , Y∗ ) → R∗ The nature of the outcomes depends on the method used for developing of M, and can include, for example, predicted response values in the case of PLS decomposition and scores and distances in the case of SVD decomposition. Using cross-validation with random splits enables the generation of a very large number of the PV-sets, which will hold the properties described above, but will be different from each other. Combining the generated PV-sets with the original training data will result in the augmented training set we are aiming at. The next two sections describe the details of PV-set generation using SVD and PLS decompositions. More information about the approach can be found in [10]. 2.1 Generation of PV-sets based on Singular Value Decomposition In the case of truncated SVD with A latent variables, the PV-model M consists of: • number of latent variables (singular vectors), A • 1 × J row-vector for mean centering, m • 1 × J row-vector for standardization (if required), s • J × A matrix with right singular vectors, V • 1 × A vector with corresponding singular values, σ. The SVD decomposition of X is defined as: Here, U is an I × A matrix with left singular vectors and Σ is an A × A diagonal matrix with singular values σ = {σ1, ..., σA}. Element E is an I × J matrix with residuals, which can be used to estimate the lack of fit both for individual data points and for the whole dataset. For the sake of simplicity we assume that columns of X are already mean centered and standardized. The right singular vectors V act as the orthonormal basis of the A-dimensional latent variable space located inside the original J-dimensional variable space (A ≤ J). The product of the left singular vectors and the singular values, T = UΣ, forms scores — coordinates of data points from X being projected to the latent variable space. The expression can also be written as: Any new data point, x = {x1, ..., xJ } can be projected to the model M by mean centering and (if required) standardizing its values using m and s, and then projecting the resulting values to the latent variable space defined by the columns of V: t = xV The score vector t can be normalized to obtain the corresponding left singular vector, u: The explained part of x can be computed as: The residual part is: And a squared Mahalanobis distance, h, between the projection of the point in the latent variable space and the origin: The first distance, q, is a measure of lack of fit for the point, while the second distance, h, is a measure of extremeness of the point (as the majority of the data points will be located around the origin due to mean centering). Hence we meet the general Procrustes rule requirement defined by Equation 4 with the outcomes R consisting of the two distances. Moreover, this rule holds true for the distances computed using any number of latent variables a = 1, ..., A. The generation of the PV-set in the case when A < rank(X) is similar but requires an additional step to hold the rule defined by Equation 14, further details can be found in [10]. Repeating the generation using random cross-validation splits provides a large number of the unique PV-sets that can be used to augment the original training set. Therefore, the SVD based data augmentation procedure has two parameters: • number of segments, K • number of latent variables, A As it will be shown in the results section, neither parameter significantly influences the quality of the augmented data. In the case of the number of latent variables, A, any number large enough for capturing the systematic variation in X, will work well. The SVD based PV-set generation does not suffer from overfitting, and optimization of A is not needed. 2.2 Generation of PV-sets based on PLS decomposition Partial least squares is a decomposition used for solving multiple linear regression problem in case when predictors (and responses if they are multiple) are collinear. The decomposition can be expressed in the following set of equations: In the case of one response, matrix Y can be replaced by a column vector y containing I response values. Then the Equation 18 can be rewritten in the following compact form: In contrast to SVD, the latent variables in PLS are oriented along directions in the column space of X, which give the largest covariance between the scores (columns of matrix T) and the values of y. Hence, by using PLS, we can prioritize the variance-covariance structure of X directly related to y. The difference between the original and estimated response values can be defined as: We can define the Procrustean rule for the regression as: Taking into account the equations above, this equation can be simplified to: or in the case of PLS: Therefore, PV-set generation in the case of PLS has the same two parameters as the SVD-based procedure: • number of segments, K • number of latent variables, A It must also be noted that the number of latent variables, A, in case of PLS must be limited (in contrast to the SVD based algorithm). If too many latent variables are selected, the variation in scalar values, ck/c, can be very large, which leads to noisy PV-sets. This happens because higher latent variables do not have any covariance with the response variable, y; hence, ck values vary chaotically. It is recommended to keep all latent variables with absolute values of ck/c within the [0, 2] range. As it is shown in the experimental part, if this limit is introduced, the number of latent variables does not have a substantial influence on the PV-sets quality and does not require specific optimization. Same can be concluded about the number of segments, K. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Sergey Kucheryavskiy, Department of Chemistry and Bioscience, Aalborg University and a Corresponding author (svk@bio.aau.dk); (2) Sergei Zhilin, CSort, LLC., Germana Titova st. 7, Barnaul, 656023, Russia and Contributing authors0 (szhilin@gmail.com). Authors: Authors: (1) Sergey Kucheryavskiy, Department of Chemistry and Bioscience, Aalborg University and a Corresponding author (svk@bio.aau.dk); (2) Sergei Zhilin, CSort, LLC., Germana Titova st. 7, Barnaul, 656023, Russia and Contributing authors0 (szhilin@gmail.com). Editor's note: This is Part 2 of 4 of a study detailing a new method for the augmentation of numeric and mixed datasets. Read the rest below. Editor's note: This is Part 2 of 4 of a study detailing a new method for the augmentation of numeric and mixed datasets. Read the rest below. Editor's note: This is Part 2 of 4 of a study detailing a new method for the augmentation of numeric and mixed datasets. Read the rest below. Editor's note: This is Part 2 of 4 of a study detailing a new method for the augmentation of numeric and mixed datasets. Read the rest below. Table of Links Abstract and 1 Introduction 2 Methods

2.1 Generation of PV-sets based on Singular Value Decomposition
2.2 Generation of PV-sets based on PLS decomposition 3 Results

3.1 Datasets
3.2 ANN regression of Tecator data
3.3 ANN classification of Heart data 4 Discussion

5 Conclusions and References Abstract and 1 Introduction Abstract and 1 Introduction 2 Methods 2.1 Generation of PV-sets based on Singular Value Decomposition
2.2 Generation of PV-sets based on PLS decomposition 2 Methods 2.1 Generation of PV-sets based on Singular Value Decomposition 2.2 Generation of PV-sets based on PLS decomposition 2.1 Generation of PV-sets based on Singular Value Decomposition 2.2 Generation of PV-sets based on PLS decomposition 3 Results 3.1 Datasets
3.2 ANN regression of Tecator data
3.3 ANN classification of Heart data 3 Results 3.1 Datasets 3.2 ANN regression of Tecator data 3.3 ANN classification of Heart data 3.1 Datasets 3.2 ANN regression of Tecator data 3.3 ANN classification of Heart data 4 Discussion 5 Conclusions and References 4 Discussion 5 Conclusions and References 5 Conclusions and References 2 Methods This chapter presents the general idea behind the Procrustes cross-validation approach, provides the necessary mathematical background and describes two implementations of the approach in detail. Let {X, Y} be two matrices of size I × J and I × M correspondingly, representing the original training set to be augmented. Columns of matrix X are predictors — variables measured or observed directly, for example, light absorbance at different wavelengths in the case of spectra, abundance of operational taxonomic units (OTU) in the case of 16S RNA sequencing, results of clinical tests for patients etc. If the original dataset has qualitative predictors, they should be replaced by corresponding dummy variables, which are combined with quantitative predictors to form X. Columns of matrix Y are responses — variables whose values should be predicted. There are situations when matrix Y is obsolete, for example, in the case of developing of one class classifier, only matrix with predictors is needed. To augment the data a specific model should be developed. It aims at capturing the variance-covariance structure of X, or its part, directly related to the variance of values in Y. This model will be referred to as the PV-model and denoted as M. The selection of the methods for creating the PV-model depends on the objectives. Thus, if data augmentation is needed for developing a one class classifier, when the matrix with responses, Y, is obsolete, we suggest using singular value decomposition (SVD), which is efficient in capturing the whole variance-covariance structure of X. In the case of augmenting data for discrimination models, there can be two solutions. One solution is to consider each class as an independent population and use SVD decomposition to capture the variance-covariance structure of X within each class. The second solution will be to create a matrix with dummy predictors, Y, where each column corresponds to a particular class, and employ PLS decomposition. More details about using SVD and PLS decompositions are provided in the next two sections. A general definition of the approach can be formulated as follows. Let us create a global PV-model M by fitting all data points from the training set {X, Y}. If this model is then applied to a new dataset, {X∗ , Y∗}, it will result in a set of outcomes R∗: M(X∗ , Y∗ ) → R∗ The nature of the outcomes depends on the method used for developing of M, and can include, for example, predicted response values in the case of PLS decomposition and scores and distances in the case of SVD decomposition. Using cross-validation with random splits enables the generation of a very large number of the PV-sets, which will hold the properties described above, but will be different from each other. Combining the generated PV-sets with the original training data will result in the augmented training set we are aiming at. The next two sections describe the details of PV-set generation using SVD and PLS decompositions. More information about the approach can be found in [10]. 2.1 Generation of PV-sets based on Singular Value Decomposition In the case of truncated SVD with A latent variables, the PV-model M consists of: • number of latent variables (singular vectors), A • 1 × J row-vector for mean centering, m • 1 × J row-vector for standardization (if required), s • J × A matrix with right singular vectors, V • 1 × A vector with corresponding singular values, σ. The SVD decomposition of X is defined as: Here, U is an I × A matrix with left singular vectors and Σ is an A × A diagonal matrix with singular values σ = {σ1, ..., σA}. Element E is an I × J matrix with residuals, which can be used to estimate the lack of fit both for individual data points and for the whole dataset. For the sake of simplicity we assume that columns of X are already mean centered and standardized. The right singular vectors V act as the orthonormal basis of the A-dimensional latent variable space located inside the original J-dimensional variable space (A ≤ J). The product of the left singular vectors and the singular values, T = UΣ, forms scores — coordinates of data points from X being projected to the latent variable space. The expression can also be written as: Any new data point, x = {x1, ..., xJ } can be projected to the model M by mean centering and (if required) standardizing its values using m and s, and then projecting the resulting values to the latent variable space defined by the columns of V: t = xV The score vector t can be normalized to obtain the corresponding left singular vector, u: The explained part of x can be computed as: The residual part is: And a squared Mahalanobis distance, h, between the projection of the point in the latent variable space and the origin: The first distance, q, is a measure of lack of fit for the point, while the second distance, h, is a measure of extremeness of the point (as the majority of the data points will be located around the origin due to mean centering). Hence we meet the general Procrustes rule requirement defined by Equation 4 with the outcomes R consisting of the two distances. Moreover, this rule holds true for the distances computed using any number of latent variables a = 1, ..., A. The generation of the PV-set in the case when A < rank(X) is similar but requires an additional step to hold the rule defined by Equation 14, further details can be found in [10]. Repeating the generation using random cross-validation splits provides a large number of the unique PV-sets that can be used to augment the original training set. Therefore, the SVD based data augmentation procedure has two parameters: • number of segments, K • number of latent variables, A As it will be shown in the results section, neither parameter significantly influences the quality of the augmented data. In the case of the number of latent variables, A, any number large enough for capturing the systematic variation in X, will work well. The SVD based PV-set generation does not suffer from overfitting, and optimization of A is not needed. 2.2 Generation of PV-sets based on PLS decomposition Partial least squares is a decomposition used for solving multiple linear regression problem in case when predictors (and responses if they are multiple) are collinear. The decomposition can be expressed in the following set of equations: In the case of one response, matrix Y can be replaced by a column vector y containing I response values. Then the Equation 18 can be rewritten in the following compact form: In contrast to SVD, the latent variables in PLS are oriented along directions in the column space of X, which give the largest covariance between the scores (columns of matrix T) and the values of y. Hence, by using PLS, we can prioritize the variance-covariance structure of X directly related to y. The difference between the original and estimated response values can be defined as: We can define the Procrustean rule for the regression as: Taking into account the equations above, this equation can be simplified to: or in the case of PLS: Therefore, PV-set generation in the case of PLS has the same two parameters as the SVD-based procedure: • number of segments, K • number of latent variables, A It must also be noted that the number of latent variables, A, in case of PLS must be limited (in contrast to the SVD based algorithm). If too many latent variables are selected, the variation in scalar values, ck/c, can be very large, which leads to noisy PV-sets. This happens because higher latent variables do not have any covariance with the response variable, y; hence, ck values vary chaotically. It is recommended to keep all latent variables with absolute values of ck/c within the [0, 2] range. As it is shown in the experimental part, if this limit is introduced, the number of latent variables does not have a substantial influence on the PV-sets quality and does not require specific optimization. Same can be concluded about the number of segments, K. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

This Smart Method Makes Small Datasets Punch Above Their Weight

Authors:
(1) Yi Ren, Tencent, Beijing, China (yiren_bj@outlook.com);
(2) Ying Du, Tencent, Beijing, China (yingdu@tencent.com);
(3) Bin Wang, Tencent, Beijing, China (hillmwang@tencent.com);
(4) Shenzheng Zhang, Tencent, Beijing, China (qjzcyzhang@tencent.com). Table of Links Abstract and 1 Introduction Methodology
Experiments
Conclusion and References ABSTRACT Recommender systems usually leverage multi-task learning methods to simultaneously optimize several objectives because of the multi-faceted user behavior data. The typical way of conducting multi-task learning(MTL) is to establish appropriate parameter sharing across multiple tasks at lower layers while reserving a separate task tower for each task at upper layers. With such design, the lower layers intend to explore the structure of task relationships and mine valuable information to be used by the task towers for accurate prediction. Since the task towers exert direct impact on the prediction results, we argue that the architecture of standalone task towers is suboptimal for promoting positive knowledge sharing. First, for each task, attending to the input information of other task towers is beneficial. For instance, the information useful for predicting the "like" task is also valuable for the "buy" task. Furthermore, because different tasks are inter-related, the training labels of multiple tasks should obey a joint distribution. It is undesirable for the prediction results for these tasks to fall into the low density areas. Accordingly, we propose the framework of Deep Mutual Learning across task towers(DML), which is compatible with various backbone multitask networks. At the entry layer of the task towers, the shared component of Cross Task Feature Mining(CTFM) is introduced to transfer input information across the task towers while still ensuring one task’s loss will not impact the inputs of other task towers. Moreover, for each task, dedicated network component called Global Knowledge Distillation(GKD) are utilized to distill valuable knowledge from the global results of the upper layer task towers to enhance the prediction consistency. Extensive offline experiments and online A/B tests are conducted to evaluate and verify the proposed approach’s effectiveness. 1. INTRODUCTION Recently, we have seen the widespread application of recommender systems, which involve different types of user feedback signals, such as clicking, rating, commenting, etc. Moreover, no single feedback signal can accurately reflect user satisfaction. For example, over-concentrating on clicking may aggravate the click-bait issue. Therefore, it is highly desirable to be able to effectively learn and estimate multiple types of user behaviors at the same time. And Multi-Task Learning is a promising technique to address this challenge. Given several related learning tasks, the goal of multi-task learning is to enhance the overall performance of different tasks by leveraging knowledge transfer among tasks. With the Multi-Task Learning(MTL) paradigm[3, 25], multiple tasks are learned simultaneously in a single model. Compared with single-task solutions, MTL costs much fewer machine resources and improves learning efficiency since we just need to train and deploy a single model. Moreover, MTL usually can warrant enhanced recommendation performance through appropriate parameter sharing. Most existing methods of Multi-Task Learning(MTL) appropriately share parameters across multiple tasks at lower layers while keeping separate task towers at upper layers. These methods can be roughly classified into four categories. The first category comprises the methods of hard parameter sharing. Among them, embedding sharing is the most intuitive structure to share information. For instance, the ESSM model [13] shares embedding parameters between the tasks of CTR (Click-Through Rate) and CVR (Conversion Rate) for improving the prediction performance of the sparse CVR task. In addition to the embedding parameters, the Shared-Bottom structure[3] is introduced to share the parameters of lower-layer MLPs among tasks. But these methods are severely plagued by the task conflicts and negative transfer issue. Second, for the methods of soft parameter sharing, each task owns separate parameters, which are regularized during training to minimize the differences between the shared parameters. L2-constrained[6] is a typical algorithm belonging to this category. Third, for the methods of customized routing, they learn customized routing weights for each task to combine and fuse information from lower-layer networks to counteract the negative transfer issue. Cross-stitch network[15] and sluice network[16] learn separate linear weights for each task to selectively merge representations from different lower-level branches. SNR[11] modularizes the shared low-level layers into parallel sub-networks and uses a transformation matrix multiplied by a scalar coding variable to learn their connections to upper-level layers to alleviate the task conflict and negative transfer issue. MSSM[5] learns differing combinations of feature fields for each expert and designs finer-grained sharing patterns among tasks through a set of coding variables that selectively choose which cells to route for a given task. But the learned routing parameters of these methods are static for all the samples, which can hardly warrant optimal performance. Finally, the methods of dynamic gating learn optimized weights for each task based on the input sample to effectively combine the outputs of lower-level networks and achieve success in industrial applications. The MMoE [12] model adapts the Mixture-of-Experts (MoE)[9] structure to multi-task learning by sharing the expert sub-networks across all tasks, while also maintaining separate gating network optimized for each task. And Zhao et al. [26] extend the MMoE model [12] and apply it to learn multiple ranking objectives in Youtube video recommender systems. Moreover, PLE [17] achieves superior performance for news recommendation by assigning both shared expert sub-networks among tasks and dedicated expert sub-networks for each task. AITM [24] is the most similar method, which also augments the architecture of task towers. Nevertheless, as a concrete implementation, it is not validated to enhance the performance of various multi-task models. Moreover, it can only work for the tasks with sequential dependence relations. Admittedly, the aforementioned methods achieve impressive performance. However, as the task towers exert a direct effect on the prediction results, the standalone task towers tend not to be the most effective design for promoting positive knowledge transfer by exploiting the task relationships. First, for each task, the information selected by the relevant tasks is extremely valuable. Accordingly, we introduce the shared component of Cross Task Feature Mining(CTFM), which utilizes delicate attention mechanisms to extract relevant information from other tasks at the entry layer of the task tower. With the common attention mechanisms, the explicit task-specific information distilled by lower-level networks are mingled together and one task’s loss will undesirably affect the inputs of other task towers, which is the task awareness missing problem and can hinder the learning of lower-level networks. In contrast to the usual attention mechanisms, our design can ensure appropriate information separation. We argue that reserving explicit task-specific knowledge has a positive effect on performance, which is validated in the experimental section. Second, because the tasks for recommender systems are related, the training labels of multiple tasks should obey a joint distribution. The prediction results for these tasks should not densely fall into the low-density areas. Therefore, a dedicated network named Global Knowledge Distillation(GKD) is introduced for each task to distill valuable global knowledge from the results of the upper layer task towers. For each task, the distilled global information helps to ensure consistent predictions with other tasks. We summarize our main contributions below. • We propose the framework of Deep Mutual Learning across task towers(DML), which is compatible with various backbone multi-task models • The proposed novel sharing structure helps to enhance effective knowledge transfer across different tasks. • We conduct offline experiments and online A/B testing to evaluate and understand the effectiveness of our method. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Yi Ren, Tencent, Beijing, China (yiren_bj@outlook.com); (2) Ying Du, Tencent, Beijing, China (yingdu@tencent.com); (3) Bin Wang, Tencent, Beijing, China (hillmwang@tencent.com); (4) Shenzheng Zhang, Tencent, Beijing, China (qjzcyzhang@tencent.com). Authors: Authors: (1) Yi Ren, Tencent, Beijing, China (yiren_bj@outlook.com); (2) Ying Du, Tencent, Beijing, China (yingdu@tencent.com); (3) Bin Wang, Tencent, Beijing, China (hillmwang@tencent.com); (4) Shenzheng Zhang, Tencent, Beijing, China (qjzcyzhang@tencent.com). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction Methodology Experiments Conclusion and References Methodology Methodology Experiments Experiments Conclusion and References Conclusion and References ABSTRACT Recommender systems usually leverage multi-task learning methods to simultaneously optimize several objectives because of the multi-faceted user behavior data. The typical way of conducting multi-task learning(MTL) is to establish appropriate parameter sharing across multiple tasks at lower layers while reserving a separate task tower for each task at upper layers. With such design, the lower layers intend to explore the structure of task relationships and mine valuable information to be used by the task towers for accurate prediction. Since the task towers exert direct impact on the prediction results, we argue that the architecture of standalone task towers is suboptimal for promoting positive knowledge sharing. First, for each task, attending to the input information of other task towers is beneficial. For instance, the information useful for predicting the "like" task is also valuable for the "buy" task. Furthermore, because different tasks are inter-related, the training labels of multiple tasks should obey a joint distribution. It is undesirable for the prediction results for these tasks to fall into the low density areas. Accordingly, we propose the framework of Deep Mutual Learning across task towers(DML), which is compatible with various backbone multitask networks. At the entry layer of the task towers, the shared component of Cross Task Feature Mining(CTFM) is introduced to transfer input information across the task towers while still ensuring one task’s loss will not impact the inputs of other task towers. Moreover, for each task, dedicated network component called Global Knowledge Distillation(GKD) are utilized to distill valuable knowledge from the global results of the upper layer task towers to enhance the prediction consistency. Extensive offline experiments and online A/B tests are conducted to evaluate and verify the proposed approach’s effectiveness. 1. INTRODUCTION Recently, we have seen the widespread application of recommender systems, which involve different types of user feedback signals, such as clicking, rating, commenting, etc. Moreover, no single feedback signal can accurately reflect user satisfaction. For example, over-concentrating on clicking may aggravate the click-bait issue. Therefore, it is highly desirable to be able to effectively learn and estimate multiple types of user behaviors at the same time. And Multi-Task Learning is a promising technique to address this challenge. Given several related learning tasks, the goal of multi-task learning is to enhance the overall performance of different tasks by leveraging knowledge transfer among tasks. With the Multi-Task Learning(MTL) paradigm[3, 25], multiple tasks are learned simultaneously in a single model. Compared with single-task solutions, MTL costs much fewer machine resources and improves learning efficiency since we just need to train and deploy a single model. Moreover, MTL usually can warrant enhanced recommendation performance through appropriate parameter sharing. Most existing methods of Multi-Task Learning(MTL) appropriately share parameters across multiple tasks at lower layers while keeping separate task towers at upper layers. These methods can be roughly classified into four categories. The first category comprises the methods of hard parameter sharing . Among them, embedding sharing is the most intuitive structure to share information. For instance, the ESSM model [13] shares embedding parameters between the tasks of CTR (Click-Through Rate) and CVR (Conversion Rate) for improving the prediction performance of the sparse CVR task. In addition to the embedding parameters, the Shared-Bottom structure[3] is introduced to share the parameters of lower-layer MLPs among tasks. But these methods are severely plagued by the task conflicts and negative transfer issue. Second, for the methods of soft parameter sharing , each task owns separate parameters, which are regularized during training to minimize the differences between the shared parameters. L2-constrained[6] is a typical algorithm belonging to this category. Third, for the methods of customized routing , they learn customized routing weights for each task to combine and fuse information from lower-layer networks to counteract the negative transfer issue. Cross-stitch network[15] and sluice network[16] learn separate linear weights for each task to selectively merge representations from different lower-level branches. SNR[11] modularizes the shared low-level layers into parallel sub-networks and uses a transformation matrix multiplied by a scalar coding variable to learn their connections to upper-level layers to alleviate the task conflict and negative transfer issue. MSSM[5] learns differing combinations of feature fields for each expert and designs finer-grained sharing patterns among tasks through a set of coding variables that selectively choose which cells to route for a given task. But the learned routing parameters of these methods are static for all the samples, which can hardly warrant optimal performance. Finally, the methods of dynamic gating learn optimized weights for each task based on the input sample to effectively combine the outputs of lower-level networks and achieve success in industrial applications. The MMoE [12] model adapts the Mixture-of-Experts (MoE)[9] structure to multi-task learning by sharing the expert sub-networks across all tasks, while also maintaining separate gating network optimized for each task. And Zhao et al. [26] extend the MMoE model [12] and apply it to learn multiple ranking objectives in Youtube video recommender systems. Moreover, PLE [17] achieves superior performance for news recommendation by assigning both shared expert sub-networks among tasks and dedicated expert sub-networks for each task. hard parameter sharing hard parameter sharing soft parameter sharing soft parameter sharing customized routing dynamic gating dynamic gating AITM [24] is the most similar method, which also augments the architecture of task towers. Nevertheless, as a concrete implementation, it is not validated to enhance the performance of various multi-task models. Moreover, it can only work for the tasks with sequential dependence relations. Admittedly, the aforementioned methods achieve impressive performance. However, as the task towers exert a direct effect on the prediction results, the standalone task towers tend not to be the most effective design for promoting positive knowledge transfer by exploiting the task relationships. First, for each task, the information selected by the relevant tasks is extremely valuable. Accordingly, we introduce the shared component of Cross Task Feature Mining(CTFM), which utilizes delicate attention mechanisms to extract relevant information from other tasks at the entry layer of the task tower. With the common attention mechanisms, the explicit task-specific information distilled by lower-level networks are mingled together and one task’s loss will undesirably affect the inputs of other task towers, which is the task awareness missing problem and can hinder the learning of lower-level networks. In contrast to the usual attention mechanisms, our design can ensure appropriate information separation. We argue that reserving explicit task-specific knowledge has a positive effect on performance, which is validated in the experimental section. Second, because the tasks for recommender systems are related, the training labels of multiple tasks should obey a joint distribution. The prediction results for these tasks should not densely fall into the low-density areas. Therefore, a dedicated network named Global Knowledge Distillation(GKD) is introduced for each task to distill valuable global knowledge from the results of the upper layer task towers. For each task, the distilled global information helps to ensure consistent predictions with other tasks. We summarize our main contributions below. • We propose the framework of D eep M utual L earning across task towers( DML ), which is compatible with various backbone multi-task models D M L DML • The proposed novel sharing structure helps to enhance effective knowledge transfer across different tasks. • We conduct offline experiments and online A/B testing to evaluate and understand the effectiveness of our method. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv

New Framework for Deep Mutual Learning (DML) to Improve Multi-task Recommender Systems

Author:
(1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Table of Links 1 Introduction 2 Dancing Links 3 The Rust Programming Language 4 RAC: Hardware/Software Co-Assurance at Scale 5 Rust and RAR 5.1 Restricted Algorithmic Rust 6 Dancing Links in Rust and 6.1 Definitions 6.2 Translation to ACL2 6.3 Dancing Links Theorems 7 Related Work 8 Conclusion 9 Acknowledgments and References 5 Rust and RAR The Rust Programming Language [16] is a modern, high-level programming language designed to combine the code generation efficiency of C/C++ with drastically improved type safety and memory management features. A distinguishing feature of Rust is that a non-scalar object may only have one owner. For example, one cannot assign a reference to an object in a local variable, and then pass that reference to a function. This restriction is similar to those imposed on ACL2 single-threaded objects (stobjs) [4], with the additional complexities of enforcing such “single-owner” restrictions in the context of a generalpurpose, imperative programming language. The Rust runtime performs array bounds checking, as well as arithmetic overflow checking (the latter can be disabled by a build environment setting). In most other ways, Rust is a fairly conventional modern programming language, with interfaces (called traits), lambdas (termed closures), and pattern matching, as well as a macro capability. Also in keeping with other modern programming language ecosystems, Rust features a language-specific build and package management sytem, named cargo. 5.1 Restricted Algorithmic Rust As we wish to utilize the RAC toolchain as a backend in our initial work, Restricted Algorithmic Rust is semantically equivalent to RAC. Thus, we adopt the same semantic restrictions as described in Russinoff’s book. Additionally, in order to enable translation to RAC, as well as to ease the transition from C/C++, RAR supports a commonly used macro that provides a C-like for loop in Rust. Note that, despite the restrictions, RAR code is proper Rust; it compiles to binary using the standard Rust compiler. RAR is transpiled to RAC via a source-to-source translator, as depicted in Fig. 3. Our transpiler is based on the plex parser and lexer generator [28] source code. We thus call our transpiler Plexi, a nickname given to a famous (and now highly sought-after) line of Marshall guitar amplifiers of the mid-1960s. Plexi performs lexical and syntactic transformations that convert RAR code to RAC code. Recent improvements in the plexi tool include better handling of array declarations, as well as providing support for Rust const declarations. The generated RAC code can then be compiled using a C/C++ compiler, fed to an HLS-based FPGA compiler, as well as translated to ACL2 via the RAC ACL2 translator, as illustrated in Fig. 3. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Author: Author: (1) David S. Hardin, Cedar Rapids, IA USA (david.s.hardin@gmail.com). Table of Links 1 Introduction 1 Introduction 2 Dancing Links 2 Dancing Links 3 The Rust Programming Language 3 The Rust Programming Language 4 RAC: Hardware/Software Co-Assurance at Scale 4 RAC: Hardware/Software Co-Assurance at Scale 5 Rust and RAR 5 Rust and RAR 5.1 Restricted Algorithmic Rust 5.1 Restricted Algorithmic Rust 6 Dancing Links in Rust and 6.1 Definitions 6 Dancing Links in Rust and 6.1 Definitions 6.2 Translation to ACL2 6.2 Translation to ACL2 6.3 Dancing Links Theorems 6.3 Dancing Links Theorems 7 Related Work 7 Related Work 8 Conclusion 8 Conclusion 9 Acknowledgments and References 9 Acknowledgments and References 5 Rust and RAR The Rust Programming Language [16] is a modern, high-level programming language designed to combine the code generation efficiency of C/C++ with drastically improved type safety and memory management features. A distinguishing feature of Rust is that a non-scalar object may only have one owner. For example, one cannot assign a reference to an object in a local variable, and then pass that reference to a function. This restriction is similar to those imposed on ACL2 single-threaded objects (stobjs) [4], with the additional complexities of enforcing such “single-owner” restrictions in the context of a generalpurpose, imperative programming language. The Rust runtime performs array bounds checking, as well as arithmetic overflow checking (the latter can be disabled by a build environment setting). In most other ways, Rust is a fairly conventional modern programming language, with interfaces (called traits), lambdas (termed closures), and pattern matching, as well as a macro capability. Also in keeping with other modern programming language ecosystems, Rust features a language-specific build and package management sytem, named cargo. 5.1 Restricted Algorithmic Rust As we wish to utilize the RAC toolchain as a backend in our initial work, Restricted Algorithmic Rust is semantically equivalent to RAC. Thus, we adopt the same semantic restrictions as described in Russinoff’s book. Additionally, in order to enable translation to RAC, as well as to ease the transition from C/C++, RAR supports a commonly used macro that provides a C-like for loop in Rust. Note that, despite the restrictions, RAR code is proper Rust; it compiles to binary using the standard Rust compiler. RAR is transpiled to RAC via a source-to-source translator, as depicted in Fig. 3. Our transpiler is based on the plex parser and lexer generator [28] source code. We thus call our transpiler Plexi, a nickname given to a famous (and now highly sought-after) line of Marshall guitar amplifiers of the mid-1960s. Plexi performs lexical and syntactic transformations that convert RAR code to RAC code. Recent improvements in the plexi tool include better handling of array declarations, as well as providing support for Rust const declarations. The generated RAC code can then be compiled using a C/C++ compiler, fed to an HLS-based FPGA compiler, as well as translated to ACL2 via the RAC ACL2 translator, as illustrated in Fig. 3. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

RRR—Rust, RAR, and RAC: The Trifecta for Co-Design Excellence

Unlocking Data Excellence: Nithin Gadicharla’s Insights into SQL Server Innovation

ডেটা এক্সেলেন্স আনলক করা: এসকিউএল সার্ভার উদ্ভাবনে নিথিন গাদিচারলার অন্তর্দৃষ্টি

Desbloqueig de l'excel·lència de dades: la visió de Nithin Gadicharla sobre la innovació de SQL Server

Ξεκλείδωμα της αριστείας δεδομένων: Οι γνώσεις του Nithin Gadicharla για την καινοτομία του SQL Server

Desbloqueo de la excelencia de los datos: perspectivas de Nithin Gadicharla sobre la innovación en SQL Server

データエクセレンスの解放: SQL Server イノベーションに関する Nithin Gadicharla の洞察

데이터 우수성 잠금 해제: SQL Server 혁신에 대한 Nithin Gadicharla의 통찰력

Duomenų tobulumo atrakinimas: Nithin Gadicharla įžvalgos apie SQL serverio naujoves

डाटा उत्कृष्टता अनलक गर्दै: एसक्यूएल सर्भर इनोभेसनमा निथिन गडिचारलाको अन्तरदृष्टि

Zhbllokimi i përsosmërisë së të dhënave: Vështrimet e Nithin Gadicharla në inovacionin e serverit SQL

Откључавање изврсности података: Увиди Нитхина Гадицхарле у иновације СКЛ сервера

Unlocking Data Excellence: Nithin Gadicharlas insikter i SQL Server Innovation

Ukuvula Ubuhle Bedatha: I-Nithin Gadicharla's Insights ku-SQL Server Innovation

Probabilistic Predictions in Classification - Evaluating Quality

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 G SOME RELATED DEFINITIONS G.1 GYROGROUPS AND GYROVECTOR SPACES Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry (Ungar, 2002; 2005; 2014). We recap the definitions of gyrogroups and gyrocommutative gyrogroups proposed in Ungar (2002; 2005; 2014). For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers. Definition G.1 (Gyrogroups (Ungar, 2014)). A pair (G, ⊕*) is a groupoid in the sense that it is a nonempty set, G, with a binary operation, ⊕. A groupoid (G,* ⊕*) is a gyrogroup if its binary operation satisfies the following axioms for a, b, c ∈ G:* (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G2) There is an element ⊖a ∈ G called a left inverse of a such that ⊖a ⊕ a = e. (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c (Left Gyroassociative Law). The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, ⊕*) is gyrocommutative if it satisfies* a ⊕ b = gyr[a, b](b ⊕ a) (Gyrocommutative Law). The following definition of gyrovector spaces is slightly different from Definition 3.2 in Ungar (2014). Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕) equipped with a scalar multiplication Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕*) equipped with a scalar multiplication* (t, x) → t ⊙ x : R × G → G is called a gyrovector space if it satisfies the following axioms for s, t ∈ R and a, b, c ∈ G*:* (V1) 1 ⊙ a = a, 0 ⊙ a = t ⊙ e = e, and (−1) ⊙ a = ⊖a. (V2) (s + t) ⊙ a = s ⊙ a ⊕ t ⊙ a. (V3) (st) ⊙ a = s ⊙ (t ⊙ a). (V4) gyr[a, b](t ⊙ c) = t ⊙ gyr[a, b]c. (V5) gyr[s ⊙ a, t ⊙ a] = Id, where Id is the identity map. G.2 AI GYROVECTOR SPACES G.3 LE GYROVECTOR SPACES G.4 LC GYROVECTOR SPACES G.5 GRASSMANN MANIFOLDS IN THE PROJECTOR PERSPECTIVE G.6 GRASSMANN MANIFOLDS IN THE ONB PERSPECTIVE G.7 THE SPD AND GRASSMANN INNER PRODUCTS G.9 THE GYRODISTANCE FUNCTION IN STRUCTURE SPACES G.10 THE PSEUDO-GYRODISTANCE FUNCTION IN STRUCTURE SPACES Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 G SOME RELATED DEFINITIONS G.1 GYROGROUPS AND GYROVECTOR SPACES Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry (Ungar, 2002; 2005; 2014). We recap the definitions of gyrogroups and gyrocommutative gyrogroups proposed in Ungar (2002; 2005; 2014). For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers. Definition G.1 (Gyrogroups (Ungar, 2014)) . A pair (G, ⊕*) is a groupoid in the sense that it is a nonempty set, G, with a binary operation, ⊕. A groupoid (G,* ⊕*) is a gyrogroup if its binary operation satisfies the following axioms for a, b, c ∈ G:* Definition G.1 (Gyrogroups (Ungar, 2014)) A pair (G, (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G2) There is an element ⊖ a ∈ G called a left inverse of a such that ⊖ a ⊕ a = e. (G2) There is an element a ∈ G called a left inverse of a such that a a = e. (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c ( Left Gyroassociative Law ). Left Gyroassociative Law The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, ⊕*) is gyrocommutative if it satisfies* Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, a ⊕ b = gyr[a, b](b ⊕ a) ( Gyrocommutative Law ). Gyrocommutative Law The following definition of gyrovector spaces is slightly different from Definition 3.2 in Ungar (2014). Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕) equipped with a scalar multiplication Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕*) equipped with a scalar multiplication* Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ( t, x ) → t ⊙ x : R × G → G t, x t x : R × G → G is called a gyrovector space if it satisfies the following axioms for s, t ∈ R and a, b, c ∈ G*:* is called a gyrovector space if it satisfies the following axioms for s, t and a, b, c (V1) 1 ⊙ a = a, 0 ⊙ a = t ⊙ e = e, and (−1) ⊙ a = ⊖a. (V1) and (V2) (s + t) ⊙ a = s ⊙ a ⊕ t ⊙ a. (V2) (V3) (st) ⊙ a = s ⊙ (t ⊙ a). (V3) (V4) gyr[a, b](t ⊙ c) = t ⊙ gyr[a, b]c. (V4) (V5) gyr[s ⊙ a, t ⊙ a] = Id, where Id is the identity map. (V5) where is the identity map. G.2 AI GYROVECTOR SPACES G.3 LE GYROVECTOR SPACES G.4 LC GYROVECTOR SPACES G.5 GRASSMANN MANIFOLDS IN THE PROJECTOR PERSPECTIVE G.6 GRASSMANN MANIFOLDS IN THE ONB PERSPECTIVE G.7 THE SPD AND GRASSMANN INNER PRODUCTS G.9 THE GYRODISTANCE FUNCTION IN STRUCTURE SPACES G.10 THE PSEUDO-GYRODISTANCE FUNCTION IN STRUCTURE SPACES Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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