Robust principal component analysis (RPCA) is a critical tool in modern machine learning, which detects outliers in the task of low-rank matrix reconstruction. In this paper, we propose a scalable and learnable non-convex approach for high-dimensional RPCA problems, which we call Learned Robust PCA (LRPCA). LRPCA is highly efficient, and its free parameters can be effectively learned to optimize via deep unfolding. Moreover, we extend deep unfolding from finite iterations to infinite iterations via a novel feedforward-recurrent-mixed neural network model. We establish the recovery guarantee of LRPCA under mild assumptions for RPCA.

The robust principal component analysis (RPCA) problem seeks to separate low-rank trends from sparse outlierswithin a data matrix, that is, to approximate a n\times d matrix D as the sum of a low-rank matrix L and a sparse matrix S .We examine the robust principal component analysis (RPCA) problem under data compression, wherethe data Y is approximately given by (L S)\cdot C, that is, a low-rank sparse data matrix that has been compressed to size n\times m (with m substantially smaller than the original dimension d) via multiplication witha compression matrix C . We give a convex program for recovering the sparse component S along with the compressed low-rank component L\cdot C, along with upper bounds on the error of this reconstructionthat scales naturally with the compression dimension m and coincides with existing results for the uncompressedsetting m d . Our results can also handle error introduced through additive noise or through missing data.The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration acrossa network of sensors, to test its performance in practice.

This paper addresses the challenge of spectral-spatial feature extraction for hyperspectral image classification by introducing a novel tensor-based framework. The proposed approach incorporates circular convolution into a tensor structure to effectively capture and integrate both spectral and spatial information. Building upon this framework, the traditional Principal Component Analysis (PCA) technique is extended to its tensor-based counterpart, referred to as Tensor Principal Component Analysis (TPCA). The proposed TPCA method leverages the inherent multi-dimensional structure of hyperspectral data, thereby enabling more effective feature representation. Experimental results on benchmark hyperspectral datasets demonstrate that classification models using TPCA features consistently outperform those using traditional PCA and other state-of-the-art techniques. These findings highlight the potential of the tensor-based framework in advancing hyperspectral image analysis.

Accurate and reliable electricity price forecasting has significant practical implications for grid management, renewable energy integration, power system planning, and price volatility management. This study focuses on enhancing electricity price forecasting in California's grid, addressing challenges from complex generation data and heteroskedasticity. Utilizing principal component analysis (PCA), we analyze CAISO's hourly electricity prices and demand from 2016-2021 to improve day-ahead forecasting accuracy. Initially, we apply traditional outlier analysis with the interquartile range method, followed by robust PCA (RPCA) for more effective outlier elimination. This approach improves data symmetry and reduces skewness. We then construct multiple linear regression models using both raw and PCA-transformed features. The model with transformed features, refined through traditional and SAS Sparse Matrix outlier removal methods, shows superior forecasting performance. The SAS Sparse Matrix method, in particular, significantly enhances model accuracy. Our findings demonstrate that PCA-based methods are key in advancing electricity price forecasting, supporting renewable integration and grid management in day-ahead markets. Keywords: Electricity price forecasting, principal component analysis (PCA), power system planning, heteroskedasticity, renewable energy integration.

The success of machine learning models relies heavily on effectively representing high-dimensional data. However, ensuring data representations capture human-understandable concepts remains difficult, often requiring the incorporation of prior knowledge and decomposition of data into multiple subspaces. Traditional linear methods fall short in modeling more than one space, while more expressive deep learning approaches lack interpretability. Here, we introduce Supervised Independent Subspace Principal Component Analysis ($\texttt{sisPCA}$), a PCA extension designed for multi-subspace learning. Leveraging the Hilbert-Schmidt Independence Criterion (HSIC), $\texttt{sisPCA}$ incorporates supervision and simultaneously ensures subspace disentanglement. We demonstrate $\texttt{sisPCA}$'s connections with autoencoders and regularized linear regression and showcase its ability to identify and separate hidden data structures through extensive applications, including breast cancer diagnosis from image features, learning aging-associated DNA methylation changes, and single-cell analysis of malaria infection. Our results reveal distinct functional pathways associated with malaria colonization, underscoring the essentiality of explainable representation in high-dimensional data analysis.

Principal component analysis (PCA) can be significantly limited when there is too few examples of the target data of interest. We propose a transfer learning approach to PCA (TL-PCA) where knowledge from a related source task is used in addition to the scarce data of a target task. Our TL-PCA has two versions, one that uses a pretrained PCA solution of the source task, and another that uses the source data. Our proposed approach extends the PCA optimization objective with a penalty on the proximity of the target subspace and the source subspace as given by the pretrained source model or the source data. This optimization is solved by eigendecomposition for which the number of data-dependent eigenvectors (i.e., principal directions of TL-PCA) is not limited to the number of target data examples, which is a root cause that limits the standard PCA performance. Accordingly, our results for image datasets show that the representation of test data is improved by TL-PCA for dimensionality reduction where the learned subspace dimension is lower or higher than the number of target data examples.

We develop two methods for the following fundamental statistical task: given an \eps -corrupted set of n samples from a d -dimensional sub-Gaussian distribution, return an approximate top eigenvector of the covariance matrix. Our first robust PCA algorithm runs in polynomial time, returns a 1 - O(\eps\log\eps {-1}) -approximate top eigenvector, and is based on a simple iterative filtering approach. Our second, which attains a slightly worse approximation factor, runs in nearly-linear time and sample complexity under a mild spectral gap assumption. These are the first polynomial-time algorithms yielding non-trivial information about the covariance of a corrupted sub-Gaussian distribution without requiring additional algebraic structure of moments. As a key technical tool, we develop the first width-independent solvers for Schatten- p norm packing semidefinite programs, giving a (1 \eps) -approximate solution in O(p\log(\tfrac{nd}{\eps})\eps {-1}) input-sparsity time iterations (where n, d are problem dimensions).

We study and provide instance-optimal algorithms in differential privacy by extending and approximating the inverse sensitivity mechanism. We provide two approximation frameworks, one which only requires knowledge of local sensitivities, and a gradient-based approximation for optimization problems, which are efficiently computable for a broad class of functions. We complement our analysis with instance-specific lower bounds for vector-valued functions, which demonstrate that our mechanisms are (nearly) instance-optimal under certain assumptions and that minimax lower bounds may not provide an accurate estimate of the hardness of a problem in general: our algorithms can significantly outperform minimax bounds for well-behaved instances. Finally, we use our approximation framework to develop private mechanisms for unbounded-range mean estimation, principal component analysis, and linear regression. For PCA, our mechanisms give an efficient (pure) differentially private algorithm with near-optimal rates.

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data.

We address the problem of denoising data from a Gaussian mixture using a two-layer non-linear autoencoder with tied weights and a skip connection. We consider the high-dimensional limit where the number of training samples and the input dimension jointly tend to infinity while the number of hidden units remains bounded. We provide closed-form expressions for the denoising mean-squared test error. Building on this result, we quantitatively characterize the advantage of the considered architecture over the autoencoder without the skip connection that relates closely to principal component analysis. We further show that our results capture accurately the learning curves on a range of real datasets.