This paper introduces the R package spca, which provides a computational framework for least squares sparse principal component analysis (LS-SPCA). Unlike other SPCA methods, LS-SPCA generates uncorrelated sparse principal components (sPCs) that effectively maximize the explained variance while maintaining strong correlations with standard principal components (PCs). The framework also includes more computationally efficient variants that produce mildly correlated sPCs, which often have lower cardinality while explaining equal or greater variance than the LS-SPCA optimal sPCs. The spca package is built on an efficient C++ backend for matrix computations, with distinct engines for tall and fat matrices, and a flexible R frontend. The user interface offers several options for computing sPCs, such as deciding whether sparsification should stop when a threshold for cumulative variance explained or R2 with the PCs is reached, and choosing between simple forward selection, stepwise forward selection, or backward elimination for variable selection. In addition to the print(), summary(), and plot() methods, the package includes tools for comparing different "spca" solutions, grouping sparse loadings, and representing foreign SPCA solutions as "spca" objects. This article demonstrates with real datasets the use of the package in a typical LS-SPCA workflow and briefly contrasts LS-SPCA with conventional SPCA solutions . Then it compares different LS-SPCA solutions obtained from the dataset. Finally, the performance of spca on large tall and fat matrices is discussed, showing that spca offers a computationally efficient alternative for computing interpretable sPCs.

Sparse Principal Component Analysis (SPCA) and Sparse Linear Regression (SLR) have a wide range of applications and have attracted a tremendous amount of attention in the last two decades as canonical examples of statistical problems in high dimension. A variety of algorithms have been proposed for both SPCA and SLR, but an explicit connection between the two had not been made. We show how to efficiently transform a black-box solver for SLR into an algorithm for SPCA: assuming the SLR solver satisfies prediction error guarantees achieved by existing efficient algorithms such as those based on the Lasso, the SPCA algorithm derived from it achieves near state of the art guarantees for testing and for support recovery for the single spiked covariance model as obtained by the current best polynomial-time algorithms. Our reduction not only highlights the inherent similarity between the two problems, but also, from a practical standpoint, allows one to obtain a collection of algorithms for SPCA directly from known algorithms for SLR. We provide experimental results on simulated data comparing our proposed framework to other algorithms for SPCA.

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SLR, but an explicit connection between the two had not been made.

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Sparse Principal Component Analysis (SPCA) and Sparse Linear Regression (SLR) have a wide range of applications and have attracted a tremendous amount of attention in the last two decades as canonical examples of statistical problems in high dimension. A variety of algorithms have been proposed for both SPCA and SLR, but an explicit connection between the two had not been made. We show how to efficiently transform a black-box solver for SLR into an algorithm for SPCA: assuming the SLR solver satisfies prediction error guarantees achieved by existing efficient algorithms such as those based on the Lasso, the SPCA algorithm derived from it achieves near state of the art guarantees for testing and for support recovery for the single spiked covariance model as obtained by the current best polynomial-time algorithms. Our reduction not only highlights the inherent similarity between the two problems, but also, from a practical standpoint, allows one to obtain a collection of algorithms for SPCA directly from known algorithms for SLR. We provide experimental results on simulated data comparing our proposed framework to other algorithms for SPCA.

Deep neural networks perform remarkably well on image classification tasks but remain vulnerable to carefully crafted adversarial perturbations. This work revisits linear dimensionality reduction as a simple, data-adapted defense. We empirically compare standard Principal Component Analysis (PCA) with its sparse variant (SPCA) as front-end feature extractors for downstream classifiers, and we complement these experiments with a theoretical analysis. On the theory side, we derive exact robustness certificates for linear heads applied to SPCA features: for both $\ell_\infty$ and $\ell_2$ threat models (binary and multiclass), the certified radius grows as the dual norms of $W^\top u$ shrink, where $W$ is the projection and $u$ the head weights. We further show that for general (non-linear) heads, sparsity reduces operator-norm bounds through a Lipschitz composition argument, predicting lower input sensitivity. Empirically, with a small non-linear network after the projection, SPCA consistently degrades more gracefully than PCA under strong white-box and black-box attacks while maintaining competitive clean accuracy. Taken together, the theory identifies the mechanism (sparser projections reduce adversarial leverage) and the experiments verify that this benefit persists beyond the linear setting. Our code is available at https://github.com/killian31/SPCARobustness.

Relation to Prior Work: On the one hand, the current text is too focused on the contributions of Purves et al. Fantastic papers of Purves et al. are very inspiring, but similar ideas were suggested before and this has to be acknowledged. Specifically, Horace Barlow proposed that matching the sensors to the statistics of the stimuli in order to reduce redundancy in the response (using a sort of linear ICA) could lead to visual illusions [Barlow90]. More generally, redundancy reduction or information maximization is connected to (nonlinear) Gaussianization and uniformization transforms. Therefore, more recent uniformization techniques such as Sequential Principal Curves Analysis (SPCA) have been proposed to explain the emergence of illusions when environment is changed [Lapàrra15]. Nonlinear transforms for error minimization [Twer01,McLeod03] may also be achieved by SPCA, thus giving alternative statistical explanation for illusions [Laparra15].

Sparse Principal Component Analysis (Sparse PCA) is a pivotal tool in data analysis and dimensionality reduction. However, Sparse PCA is a challenging problem in both theory and practice: it is known to be NP-hard and current exact methods generally require exponential runtime. In this paper, we propose a novel framework to efficiently approximate Sparse PCA by (i) approximating the general input covariance matrix with a re-sorted block-diagonal matrix, (ii) solving the Sparse PCA sub-problem in each block, and (iii) reconstructing the solution to the original problem. Our framework is simple and powerful: it can leverage any off-the-shelf Sparse PCA algorithm and achieve significant computational speedups, with a minor additive error that is linear in the approximation error of the block-diagonal matrix. Suppose $g(k, d)$ is the runtime of an algorithm (approximately) solving Sparse PCA in dimension $d$ and with sparsity value $k$. Our framework, when integrated with this algorithm, reduces the runtime to $\mathcal{O}\left(\frac{d}{d^\star} \cdot g(k, d^\star) + d^2\right)$, where $d^\star \leq d$ is the largest block size of the block-diagonal matrix. For instance, integrating our framework with the Branch-and-Bound algorithm reduces the complexity from $g(k, d) = \mathcal{O}(k^3\cdot d^k)$ to $\mathcal{O}(k^3\cdot d \cdot (d^\star)^{k-1})$, demonstrating exponential speedups if $d^\star$ is small. We perform large-scale evaluations on many real-world datasets: for exact Sparse PCA algorithm, our method achieves an average speedup factor of 93.77, while maintaining an average approximation error of 2.15%; for approximate Sparse PCA algorithm, our method achieves an average speedup factor of 6.77 and an average approximation error of merely 0.37%.

The authors consider the problem of large-scale GP regression; they propose a multiresolution approximation method for the Gram matrix K. In the literature, most approximation approaches assume either (1) a low rank representation for K, which may not be supported by the data, or (2) a block-diagonal form for K, the structure of which has to be identified by clustering methods, which is not trivial for high-dimensional data. The current paper proposes MKA, a novel approximation approach that uses captures local and global properties for K. The Gram matrix K is approximated as a Kronecker sum of low-rank and diagonal matrices, a fact that significantly reduces the computational complexity of the linear algebra calculations required in the context of GP regression. The paper initiates a very interesting discussion on the nature of local and global kernel approximations, but I feel that certain aspects ofthe methodology proposed are not sufficiently clear.