Nonlinear component analysis such as kernel Principle Component Analysis (KPCA) and kernel Canonical Correlation Analysis (KCCA) are widely used in machine learning, statistics and data analysis, but they can not scale up to big datasets. Recent attempts have employed random feature approximations to convert the problem to the primal form for linear computational complexity. However, to obtain high quality solutions, the number of random features should be the same order of magnitude as the number of data points, making such approach not directly applicable to the regime with millions of data points. We propose a simple, computationally efficient, and memory friendly algorithm based on the "doubly stochastic gradients" to scale up a range of kernel nonlinear component analysis, such as kernel PCA, CCA and SVD. Despite the nature of these problems, our method enjoys theoretical guarantees that it converges at the rate @math to the global optimum, even for the top @math eigen subspace. Unlike many alternatives, our algorithm does not require explicit orthogonalization, which is infeasible on big datasets. We demonstrate the effectiveness and scalability of our algorithm on large scale synthetic and real world datasets.
For finite time convergence rate, @cite_10 proved the @math rate for the top eigenvector in linear PCA using Oja's rule. For the same task, @cite_15 proposed a noise reduced PCA with linear convergence rate, where the rate is in terms of epochs, , number of passes over the whole dataset. The noisy power method presented in @cite_27 provided linear convergence for a subspace, although it only converges linearly to a constant error level. In addition, the updates require explicit orthogonalization, which is impractical for kernel methods. In comparison, our method converges in @math for a subspace, without the need for orthogonalization.
@cite_27 @cite_15 @cite_10
2166581276 2952199998 1988351624 2951713178 2099953425 2277079409 2034171981 2246636461 2142304170 2143894665 2963297691 2101780216 2037549374 2592062427 2736399023 2136860609 2156204103 2950132609 2116195245 2155723880
We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications: Streaming PCA. A recent work of (NIPS 2013) gives a space-efficient algorithm for PCA in a streaming model where samples are drawn from a gaussian spiked covariance model. We give a simpler and more general analysis that applies to arbitrary distributions confirming experimental evidence of Moreover, even in the spiked covariance model our result gives quantitative improvements in a natural parameter regime. It is also notably simpler and follows easily from our general convergence analysis of the noisy power method together with a matrix Chernoff bound. Private PCA. We provide the first nearly-linear time algorithm for the problem of differentially private principal component analysis that achieves nearly tight worst-case error bounds. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by an essentially tight dependence on the coherence of the matrix. This result resolves the main problem left open by Hardt and Roth (STOC 2013). The coherence is always bounded by the matrix dimension but often substantially smaller thus leading to strong average-case improvements over the optimal worst-case bound. We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications including streaming PCA and privacy-preserving singular vector computation. We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. We propose a new method for robust PCA -- the task of recovering a low-rank matrix from sparse corruptions that are of unknown value and support. Our method involves alternating between projecting appropriate residuals onto the set of low-rank matrices, and the set of sparse matrices; each projection is non-convex but easy to compute. In spite of this non-convexity, we establish exact recovery of the low-rank matrix, under the same conditions that are required by existing methods (which are based on convex optimization). For an @math input matrix ( @math , our method has a running time of @math per iteration, and needs @math iterations to reach an accuracy of @math . This is close to the running time of simple PCA via the power method, which requires @math per iteration, and @math iterations. In contrast, existing methods for robust PCA, which are based on convex optimization, have @math complexity per iteration, and take @math iterations, i.e., exponentially more iterations for the same accuracy. Experiments on both synthetic and real data establishes the improved speed and accuracy of our method over existing convex implementations. We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose a fast iterative algorithm, which linearly converges to the matrix minimizing the convex problem. We also quantify the effect of noise and regularization and discuss many other practical and theoretical issues for improving the subspace recovery in various settings. When replacing the sum of terms in the convex energy function (that we minimize) with the sum of squares of terms, we obtain that the new minimizer is a scaled version of the inverse sample covariance (when exists). We thus interpret our minimizer and its subspace (spanned by its bottom eigenvectors) as robust versions of the empirical inverse covariance and the PCA subspace respectively. We compare our method with many other algorithms for robust PCA on synthetic and real data sets and demonstrate state-of-the-art speed and accuracy. We consider the noisy power method algorithm, which has wide applications in machine learning and statistics, especially those related to principal component analysis (PCA) under resource (communication, memory or privacy) constraints. Existing analysis of the noisy power method shows an unsatisfactory dependency over the "consecutive" spectral gap @math of an input data matrix, which could be very small and hence limits the algorithm's applicability. In this paper, we present a new analysis of the noisy power method that achieves improved gap dependency for both sample complexity and noise tolerance bounds. More specifically, we improve the dependency over @math to dependency over @math , where @math is an intermediate algorithm parameter and could be much larger than the target rank @math . Our proofs are built upon a novel characterization of proximity between two subspaces that differ from canonical angle characterizations analyzed in previous works. Finally, we apply our improved bounds to distributed private PCA and memory-efficient streaming PCA and obtain bounds that are superior to existing results in the literature. This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems. We study the problem of recovering the subspace spanned by the first @math principal components of @math -dimensional data under the streaming setting, with a memory bound of @math . Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheless, the convergence rate of the family can be seriously affected by the learning rate of the descent steps and deserves more serious study. The second family is based on the power method over blocks of data, but setting the block size for its existing algorithms is not an easy task. In this paper, we analyze the convergence rate of a representative algorithm with decayed learning rate (Oja and Karhunen, 1985) in the first family for the general @math case. Moreover, we propose a novel algorithm for the second family that sets the block sizes automatically and dynamically with faster convergence rate. We then conduct empirical studies that fairly compare the two families on real-world data. The studies reveal the advantages and disadvantages of these two families. We introduce two methods to improve convergence of the Kernel Hebbian Algorithm (KHA) for iterative kernel PCA. KHA has a scalar gain parameter which is either held constant or decreased as 1 t, leading to slow convergence. Our KHA et algorithm accelerates KHA by incorporating the reciprocal of the current estimated eigenvalues as a gain vector. We then derive and apply Stochastic Meta-Descent (SMD) to KHA et; this further speeds convergence by performing gain adaptation in RKHS. Experimental results for kernel PCA and spectral clustering of USPS digits as well as motion capture and image de-noising problems confirm that our methods converge substantially faster than conventional KHA. We derive the l! convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two di! erent assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small. We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at O(T-3 4), while the sum of utilities converges to an approximate optimum at O(T-1)-an improvement upon the worst case O(T-1 2) rates. We show a black-box reduction for any algorithm in the class to achieve O(T-1 2) rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of Rakhlin and Shridharan [17] and [4], who only analyzed two-player zero-sum games for specific algorithms. In this paper, we study the estimation of the k-dimensional sparse principal sub-space of covariance matrix Σ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-k, and attains a √s n statistical rate of convergence with s being the subspace sparsity level and n the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets. We consider the problem of fitting one or more subspaces to a collection of data points drawn from the subspaces and corrupted by noise outliers. We pose this problem as a rank minimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean, self-expressive, low-rank dictionary plus a matrix of noise outliers. Our key contribution is to show that, for noisy data, this non-convex problem can be solved very efficiently and in closed form from the SVD of the noisy data matrix. Remarkably, this is true for both one or more subspaces. An important difference with respect to existing methods is that our framework results in a polynomial thresholding of the singular values with minimal shrinkage. Indeed, a particular case of our framework in the case of a single subspace leads to classical PCA, which requires no shrinkage. In the case of multiple subspaces, our framework provides an affinity matrix that can be used to cluster the data according to the sub-spaces. In the case of data corrupted by outliers, a closed-form solution appears elusive. We thus use an augmented Lagrangian optimization framework, which requires a combination of our proposed polynomial thresholding operator with the more traditional shrinkage-thresholding operator. In this paper we employ a parallel version of a randomized (block) coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable nonsmooth convex function. Under the assumption of Lipschitz continuity of the gradient of the smooth function, this method has a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the smooth component of the objective function. Spectral clustering based subspace clustering methods have emerged recently. When the inputs are 2-dimensional (2D) data, most existing clustering methods convert such data to vectors as preprocessing, which severely damages spatial information of the data. In this paper, we propose a novel subspace clustering method for 2D data with enhanced capability of retaining spatial information for clustering. It seeks two projection matrices and simultaneously constructs a linear representation of the projected data, such that the sought projections help construct the most expressive representation with the most variational information. We regularize our method based on covariance matrices directly obtained from 2D data, which have much smaller size and are more computationally amiable. Moreover, to exploit nonlinear structures of the data, a nonlinear version is proposed, which constructs an adaptive manifold according to updated projections. The learning processes of projections, representation, and manifold thus mutually enhance each other, leading to a powerful data representation. Efficient optimization procedures are proposed, which generate non-increasing objective value sequence with theoretical convergence guarantee. Extensive experimental results confirm the effectiveness of proposed method. Linear subspace methods that provide sufficient reconstruction of the data, such as PCA, offer an efficient way of dealing with missing pixels, outliers, and occlusions that often appear in the visual data. Discriminative methods, such as LDA, which, on the other hand, are better suited for classification tasks, are highly sensitive to corrupted data. We present a theoretical framework for achieving the best of both types of methods: an approach that combines the discrimination power of discriminative methods with the reconstruction property of reconstructive methods which enables one to work on subsets of pixels in images to efficiently detect and reject the outliers. The proposed approach is therefore capable of robust classification with a high-breakdown point. We also show that subspace methods, such as CCA, which are used for solving regression tasks, can be treated in a similar manner. The theoretical results are demonstrated on several computer vision tasks showing that the proposed approach significantly outperforms the standard discriminative methods in the case of missing pixels and images containing occlusions and outliers. While kernel canonical correlation analysis (CCA) has been applied in many contexts, the convergence of finite sample estimates of the associated functions to their population counterparts has not yet been established. This paper gives a mathematical proof of the statistical convergence of kernel CCA, providing a theoretical justification for the method. The proof uses covariance operators defined on reproducing kernel Hilbert spaces, and analyzes the convergence of their empirical estimates of finite rank to their population counterparts, which can have infinite rank. The result also gives a sufficient condition for convergence on the regularization coefficient involved in kernel CCA: this should decrease as n-1 3, where n is the number of data. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning @math (where @math is a bound on the smoothness and @math on the strong convexity) to a linear dependence on @math . Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate. We consider a situation in which we see samples in @math drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both. The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A-ND algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A-NC algorithm where the weights are constant but consensus is run for a fixed number of iterations [^(iota)], then it is restarted and rerun for a total of [^(p)] runs, and at the end averages the final states of the [^(p)] runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A-ND to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that A-ND represents the best of both worlds-zero bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A-NC, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations [^(iota)][^(p)] , shorter runs (smaller [^(iota)] ) lead to high bias but smaller variance (larger number [^(p)] of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for A-NC to reach in the shortest number of iterations [^(iota)][^(p)] , with high probability (1-delta), (epsiv, delta)-consensus (epsiv residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.