Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional feature space defined by a kernel function, i.e., a function returning the inner product between the images of two data points in the feature space. Central to any kernel method is the kernel matrix, which is built by evaluating the kernel function on a given sample dataset. In this paper, we initiate the study of non-asymptotic spectral theory of random kernel matrices. These are n x n random matrices whose (i,j)th entry is obtained by evaluating the kernel function on @math and @math , where @math are a set of n independent random high-dimensional vectors. Our main contribution is to obtain tight upper bounds on the spectral norm (largest eigenvalue) of random kernel matrices constructed by commonly used kernel functions based on polynomials and Gaussian radial basis. As an application of these results, we provide lower bounds on the distortion needed for releasing the coefficients of kernel ridge regression under attribute privacy, a general privacy notion which captures a large class of privacy definitions. Kernel ridge regression is standard method for performing non-parametric regression that regularly outperforms traditional regression approaches in various domains. Our privacy distortion lower bounds are the first for any kernel technique, and our analysis assumes realistic scenarios for the input, unlike all previous lower bounds for other release problems which only hold under very restrictive input settings.
Like the development of non-asymptotic theory of traditional random matrices has found multitude of applications in areas including statistics, geometric functional analysis, and compressed sensing @cite_13 , we believe that the growth of a non-asymptotic theory of random kernel matrices will help in better understanding of many machine learning applications that utilize kernel techniques.
@cite_13
2131172946
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists. The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value). In the last few years, the asymptotic distribution of the singular values of certain random matrices has emerged as a key tool in the analysis and design of wireless communication channels. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual application. The goal of this paper is the investigation and application of random matrix theory with particular emphasis on the asymptotic theorems on the distribution of the squared singular values under various assumption on the joint distribution of the random matrix entries. Neural network configurations with random weights play an important role in the analysis of deep learning. They define the initial loss landscape and are closely related to kernel and random feature methods. Despite the fact that these networks are built out of random matrices, the vast and powerful machinery of random matrix theory has so far found limited success in studying them. A main obstacle in this direction is that neural networks are nonlinear, which prevents the straightforward utilization of many of the existing mathematical results. In this work, we open the door for direct applications of random matrix theory to deep learning by demonstrating that the pointwise nonlinearities typically applied in neural networks can be incorporated into a standard method of proof in random matrix theory known as the moments method. The test case for our study is the Gram matrix @math , @math , where @math is a random weight matrix, @math is a random data matrix, and @math is a pointwise nonlinear activation function. We derive an explicit representation for the trace of the resolvent of this matrix, which defines its limiting spectral distribution. We apply these results to the computation of the asymptotic performance of single-layer random feature methods on a memorization task and to the analysis of the eigenvalues of the data covariance matrix as it propagates through a neural network. As a byproduct of our analysis, we identify an intriguing new class of activation functions with favorable properties. Hoeffding's U-statistics model combinatorial-type matrix parameters (appearing in CS theory) in a natural way. This paper proposes using these statistics for analyzing random compressed sensing matrices, in the non-asymptotic regime (relevant to practice). The aim is to address certain pessimisms of worst-case restricted isometry analyses, as observed by both Blanchard and Dossal, We show how U-statistics can obtain average-case analyses, by relating to statistical restricted isometry property (StRIP) type recovery guarantees. However unlike standard StRIP, random signal models are not required; the analysis used here holds in the almost sure (probabilistic) sense. For Gaussian bounded entry matrices, we show that both @math -minimization and LASSO essentially require on the order of @math measurements to respectively recover at least @math fraction, and @math fraction, of the signals. Noisy conditions are considered. Empirical evidence suggests our analysis to compare well to Donoho and Tanner's recent large deviation bounds for @math -equivalence, in the regime of block lengths @math with high undersampling ( @math measurements); similar system sizes are found in recent CS implementation. In this work, it is assumed throughout that matrix columns are independently sampled. In order to fully utilize "big data", it is often required to use "big models". Such models tend to grow with the complexity and size of the training data, and do not make strong parametric assumptions upfront on the nature of the underlying statistical dependencies. Kernel methods fit this need well, as they constitute a versatile and principled statistical methodology for solving a wide range of non-parametric modelling problems. However, their high computational costs (in storage and time) pose a significant barrier to their widespread adoption in big data applications. We propose an algorithmic framework and high-performance implementation for massive-scale training of kernel-based statistical models, based on combining two key technical ingredients: (i) distributed general purpose convex optimization, and (ii) the use of randomization to improve the scalability of kernel methods. Our approach is based on a block-splitting variant of the Alternating Directions Method of Multipliers, carefully reconfigured to handle very large random feature matrices, while exploiting hybrid parallelism typically found in modern clusters of multicore machines. Our implementation supports a variety of statistical learning tasks by enabling several loss functions, regularization schemes, kernels, and layers of randomized approximations for both dense and sparse datasets, in a highly extensible framework. We evaluate the ability of our framework to learn models on data from applications, and provide a comparison against existing sequential and parallel libraries. We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs @math converging locally to a Galton--Watson tree @math (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function @math of @math . In the first part, we show that the adjacency operator associated with @math is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on @math for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of @math . Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics. Given a high-dimensional data matrix @math , Approximate Message Passing (AMP) algorithms construct sequences of vectors @math , @math , indexed by @math by iteratively applying @math or @math , and suitable non-linear functions, which depend on the specific application. Special instances of this approach have been developed --among other applications-- for compressed sensing reconstruction, robust regression, Bayesian estimation, low-rank matrix recovery, phase retrieval, and community detection in graphs. For certain classes of random matrices @math , AMP admits an asymptotically exact description in the high-dimensional limit @math , which goes under the name of state evolution.' Earlier work established state evolution for separable non-linearities (under certain regularity conditions). Nevertheless, empirical work demonstrated several important applications that require non-separable functions. In this paper we generalize state evolution to Lipschitz continuous non-separable nonlinearities, for Gaussian matrices @math . Our proof makes use of Bolthausen's conditioning technique along with several approximation arguments. In particular, we introduce a modified algorithm (called LAMP for Long AMP) which is of independent interest. Random matrices now play a role in many areas of theoretical, applied,and computational mathematics. Therefore, it is desirable to have toolsfor studying random matrices that are flexible, easy to use, and powerful.Over the last fifteen years, researchers have developed a remarkablefamily of results, called matrix concentration inequalities, that achieveall of these goals.This monograph offers an invitation to the field of matrix concentrationinequalities. It begins with some history of random matrix theory;it describes a flexible model for random matrices that is suitablefor many problems; and it discusses the most important matrix concentrationresults. To demonstrate the value of these techniques, thepresentation includes examples drawn from statistics, machine learning,optimization, combinatorics, algorithms, scientific computing, andbeyond. This paper establishes information-theoretic limits for estimating a finite-field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse-a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n × n-sensing matrices contain, on average, Ω(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the aforementioned results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive minimum distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth. Machine generated contents note: 1. Introduction to compressed sensing Mark A. Davenport, Marco F. Duarte, Yonina C. Eldar and Gitta Kutyniok; 2. Second generation sparse modeling: structured and collaborative signal analysis Alexey Castrodad, Ignacio Ramirez, Guillermo Sapiro, Pablo Sprechmann and Guoshen Yu; 3. Xampling: compressed sensing of analog signals Moshe Mishali and Yonina C. Eldar; 4. Sampling at the rate of innovation: theory and applications Jose Antonia Uriguen, Yonina C. Eldar, Pier Luigi Dragotta and Zvika Ben-Haim; 5. Introduction to the non-asymptotic analysis of random matrices Roman Vershynin; 6. Adaptive sensing for sparse recovery Jarvis Haupt and Robert Nowak; 7. Fundamental thresholds in compressed sensing: a high-dimensional geometry approach Weiyu Xu and Babak Hassibi; 8. Greedy algorithms for compressed sensing Thomas Blumensath, Michael E. Davies and Gabriel Rilling; 9. Graphical models concepts in compressed sensing Andrea Montanari; 10. Finding needles in compressed haystacks Robert Calderbank, Sina Jafarpour and Jeremy Kent; 11. Data separation by sparse representations Gitta Kutyniok; 12. Face recognition by sparse representation Arvind Ganesh, Andrew Wagner, Zihan Zhou, Allen Y. Yang, Yi Ma and John Wright. A standard Gelfand–Tsetlin pattern of depth n is a configuration of particles in ( 1, ,n R ) . For each ( r 1, , n , r R ) is referred to as the rth level of the pattern. A standard Gelfand–Tsetlin pattern has exactly r particles on each level r, and particles on adjacent levels satisfy an interlacing constraint. Probability distributions on the set of Gelfand–Tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random Hermitian matrices of size n. We consider such probability spaces when the distribution of the matrix is unitarily invariant, prove a determinantal structure for a broad subclass, and calculate the correlation kernel. In particular we consider the case where the eigenvalues of the random matrix are fixed. This corresponds to choosing uniformly from the set of Gelfand–Tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix. Fixing ( q_n 1, ,n ) , and letting n → ∞ under the assumption that ( q_n n (0, 1) ) and the empirical distribution of the particles on the nth level converges weakly, the asymptotic behaviour of particles on level q n is relevant to free probability theory. Saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the Sine kernel. Random projection is a simple technique that has had a number of applications in algorithm design. In the context of machine learning, it can provide insight into questions such as “why is a learning problem easier if data is separable by a large margin?” and “in what sense is choosing a kernel much like choosing a set of features?” This talk is intended to provide an introduction to random projection and to survey some simple learning algorithms and other applications to learning based on it. I will also discuss how, given a kernel as a black-box function, we can use various forms of random projection to extract an explicit small feature space that captures much of what the kernel is doing. This talk is based in large part on work in [BB05, BBV04] joint with Nina Balcan and Santosh Vempala. Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that storing and computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices, when combined with diagonal Gaussian matrices, exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks proposed by Rahimi and Recht (2009) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy. Our method applies to any translation invariant and any dot-product kernel, such as the popular RBF kernels and polynomial kernels. We prove that the approximation is unbiased and has low variance. Experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and or require real-time prediction. This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider @math with @math where @math and @math are random rectangular matrices with i.i.d. centered entries. The function @math is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where @math and @math have sub-Gaussian tails and @math is real analytic. This extends a previous result where the case of Gaussian matrices @math and @math is considered. We also investigate the same questions in the multi-layer case, regarding neural network applications. We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two dimensions. The algorithm is ''fast'' in the sense that its asymptotic complexity is O(n), where n is the number of nodes in the discretization. Unlike previous fast techniques based on iterative solvers, the present algorithm directly constructs a compressed factorization of the inverse of the matrix; thus it is suitable for problems involving relatively ill-conditioned matrices, and is particularly efficient in situations involving multiple right hand sides. The performance of the scheme is illustrated with several numerical examples. rformance of the scheme is illustrated with several numerical examples. ples. The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices. Because of the substantial computational cost due to lifting, many approaches to phase retrieval have been developed, including non-convex optimization algorithms which operate in the natural parameter space, such as Wirtinger Flow. Very recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under optimal sample complexity. The current proofs of PhaseMax rely on statistical learning theory or geometric probability theory. Here, we present a short and elementary proof that PhaseMax exactly recovers real-valued vectors from random measurements under optimal sample complexity. Our proof only relies on standard probabilistic concentration and covering arguments, yielding a simpler and more direct proof than those that require statistical learning theory, geometric probability or the highly technical arguments for Wirtinger Flow-like approaches. From the Publisher: In the 1990s, a new type of learning algorithm was developed, based on results from statistical learning theory: the Support Vector Machine (SVM). This gave rise to a new class of theoretically elegant learning machines that use a central concept of SVMs-kernels--for a number of learning tasks. Kernel machines provide a modular framework that can be adapted to different tasks and domains by the choice of the kernel function and the base algorithm. They are replacing neural networks in a variety of fields, including engineering, information retrieval, and bioinformatics. Learning with Kernels provides an introduction to SVMs and related kernel methods. Although the book begins with the basics, it also includes the latest research. It provides all of the concepts necessary to enable a reader equipped with some basic mathematical knowledge to enter the world of machine learning using theoretically well-founded yet easy-to-use kernel algorithms and to understand and apply the powerful algorithms that have been developed over the last few years. Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices when combined with diagonal Gaussian matrices exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks (Rahimi & Recht, 2007) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy. We prove that the approximation is unbiased and has low variance. Extensive experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and or require real-time prediction.