Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Principal Component Analysis (PCA) is a widely utilized technique for dimensionality reduction; however, its inherent lack of interpretability-stemming from dense linear combinations of all feature-limits its applicability in many domains. In this paper, we propose a novel sparse PCA algorithm that imposes sparsity through a smooth L1 penalty and leverages a Hamiltonian formulation solved via geometric integration techniques. Specifically, we implement two distinct numerical methods-one based on the Proximal Gradient (ISTA) approach and another employing a leapfrog (fourth-order Runge-Kutta) scheme-to minimize the energy function that balances variance maximization with sparsity enforcement. To extract a subset of sparse principal components, we further incorporate a deflation technique and subsequently transform the original high-dimensional face data into a lower-dimensional feature space. Experimental evaluations on a face recognition dataset-using both k-nearest neighbor and kernel ridge regression classifiers-demonstrate that the proposed sparse PCA methods consistently achieve higher classification accuracy than conventional PCA. Future research will extend this framework to integrate sparse PCA with modern deep learning architectures for multimodal recognition tasks.

This paper concerns the problem of estimating a vector \beta, from /-1 measurements y_i which depend statistically on linear functions x_i, \beta, where the x_i are Gaussian random vectors. This model is general enough to capture compressed sensing and phase retrieval problems with binary measurements. The paper assumes that f is completely unknown ahead of time, but that it satisfies certain moment conditions. The paper shows how, under these moment conditions, to reduce the problem of estimating \beta to a sparse PCA problem, with covariance matrix generated by pairs of observations. The idea is to look at differences of x_i - x_{i'} and y_i - y_{i'}; the paper proves that the population covariance matrix of \delta_y \delta_x is a spiked identity matrix, where the spike is of the form \beta* \beta* T. This clever reduction appears to be the main contribution of the work.

We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch.

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. Such components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the input data. We evaluate our algorithm on real datasets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

We consider the following multi-component sparse PCA problem:given a set of data points, we seek to extract a small number of sparse components with \emph{disjoint} supports that jointly capture the maximum possible variance.Such components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but this greedy procedure is suboptimal.We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to $1$ from the optimal.Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem.Its complexity grows as a low order polynomial in the ambient dimension of the input data, but exponentially in its rank.However, it can be effectively applied on a low-dimensional sketch of the input data.We evaluate our algorithm on real datasets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

We study how well one can recover sparse principal componentsof a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems,if the sketch is close (in the spectral norm) to the original datamatrix, then one can recover a near optimal solution to the optimizationproblem by using the sketch. In particular, we use this approach toobtain sparse principal components and show that for \math{m} data pointsin \math{n} dimensions,\math{O(\epsilon^{-2}\tilde k\max\{m,n\})} elements gives an\math{\epsilon}-additive approximation to the sparse PCA problem(\math{\tilde k} is the stable rank of the data matrix).We demonstrate our algorithms extensivelyon image, text, biological and financial data.The results show that not only are we able to recover the sparse PCAs from the incomplete data, but by using our sparse sketch, the running timedrops by a factor of five or more.

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch. In particular, we use this approach to obtain sparse principal components and show that for \math{m} data points in \math{n} dimensions, \math{O(\epsilon^{-2}\tilde k\max\{m,n\})} elements gives an \math{\epsilon}-additive approximation to the sparse PCA problem (\math{\tilde k} is the stable rank of the data matrix). We demonstrate our algorithms extensively on image, text, biological and financial data. The results show that not only are we able to recover the sparse PCAs from the incomplete data, but by using our sparse sketch, the running time drops by a factor of five or more.