DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.
problem Efficiently decomposing tensors with deep learning to capture nonlinear structures.
method Low-rank tensor decomposition using deep generative networks trained to minimize approximation error.
result DeepTensor outperforms classical methods like SVD and PCA in various applications, including image denoising and 3D MRI.
A simple self-supervised model for tensor RPCA using deep unfolding.
problem Tensor robust principal component analysis (RPCA) challenges in practical applications.
method Deep unfolding with only four hyperparameters.
result Competitive or superior performance compared to supervised methods, even in data-starved scenarios.
Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the resid…
Paper studies statistical-computational trade-offs in tensor PCA and related problems.
problem Statistical-computational gap in tensor PCA estimation.
method Derives computational lower bounds using communication complexity.
result Lower bounds specify trade-off among passes, sample size, and memory.
Paper proposes a new method for exact recovery in robust tensor principal component analysis.
problem Exact recovery of low-rank and sparse components in tensors.
method Proposes a new method based on tensor-tensor product and t-SVD to solve a convex optimization problem.
result Exact recovery achieved in a deterministic fashion without randomness assumptions.
Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…
Robust TOT regression method handles outliers in tensor data.
problem Outliers in tensor data affect standard TOT regression.
method ROTOT method using a single loss function for outliers and robust MPCA for predictor.
result ROTOT method reduces influence of both casewise and cellwise outliers.
Efficient algorithm for tensor PCA with improved time complexity.
problem Tensor PCA problem with signal-to-noise ratio constraint.
method Counting specific weighted hypergraphs to solve tensor PCA.
result Algorithm runs in time nC+o(1) for λn−4p signal-to-noise ratio. Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-1 decompositions. Our main appli…
New robust MPCA method handles casewise and cellwise outliers in tensor data.
problem Outliers, especially casewise and cellwise, affect the performance of standard MPCA.
method Uses a single loss function to reduce the influence of both types of outliers and missing values.
result The new method improves robustness and performance in tensor data analysis.
Unified framework improves PCA for outliers and distributed data.
problem Outliers and limitations in PCA for large-scale applications.
method φ-PCA framework that retains PCA efficiency and adds robustness.
result HM-PCA achieves optimal robustness and efficiency.
New algorithms recover sparse tensor principal components efficiently.
problem Recovering sparse tensor principal components from noisy data.
method Family of algorithms interpolating between polynomial-time and exhaustive search, tailored for sparse and highly sparse regimes.
result Our algorithms recover sparse vectors for signal-to-noise ratios beyond previous limits, with time complexity ildeO(np+t). Face recognition is the important field in machine learning and pattern recognition research area. It has a lot of applications in military, finance, public security, to name a few. In this paper, the combination of the tensor sparse PCA with the nearest-neighbor method (and with the kernel ridge regression method) wil…
Study of Langevin dynamics for tensor PCA recovery in high dimensions.
problem Recovering hidden signal vectors (spikes) from noisy Gaussian tensor observations.
method Langevin dynamics approach for nonconvex optimization.
result Sample complexity matches the single-spike case but degrades for all spikes.
The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2…
We develop efficient algorithms for robust PCA that handle outliers.
problem Finding principal components in datasets with outliers.
method Nearly-linear time and streaming algorithms for robust PCA.
result Near-optimal error guarantees for robust PCA with nearly-linear time and memory usage.
Tensor PCA problem analyzed with statistical query lower bounds.
problem Estimating the expected value of a rank-1 tensor from Gaussian samples.
method Sharp analysis of optimal sample complexity in the Statistical Query model.
result SQ algorithms with polynomial query complexity fail in the conjectured hard phase and have sub-optimal sample complexity.
New algorithms recover signals robustly against outliers and heavy-tailed noise.
problem Estimating signals in the presence of heavy-tailed noise and outliers.
method Sum-of-squares relaxations for robust estimation.
result Recovery of signals in polynomial/quasipolynomial time for specific problems.
Sharp analysis of power iteration for tensor PCA, improving convergence and stopping criteria.
problem Analyzing the power iteration algorithm for tensor PCA to improve convergence and stopping criteria.
method Sharp bounds on the number of iterations, revealing a smaller algorithmic threshold, proposing a stopping criterion.
result Sharp bounds on the number of iterations required for power method to converge, revealing a smaller algorithmic threshold than previously conjectured.
Paper develops inference methods for low-rank tensors without debiasing.
problem Statistical inference for low-rank tensor models.
method Two-iteration alternating minimization for asymptotic distribution.
result Asymptotic distributions and confidence regions for singular subspaces.
PCA++ improves robustness to background noise in contrastive learning.
problem Recovering shared signal subspaces from positive pairs in high-dimensional data with structured background noise.
method PCA++ uses hard uniformity-constrained contrastive learning to enforce identity covariance on projected features.
result PCA++ outperforms standard PCA and alignment-only PCA+ in simulations and real-world datasets.
Optimal tensor PCA for estimating factors and loadings in high-dimensional panel data.
problem Estimating factors and loadings in high-dimensional panel data with non-negligible correlations.
method Tensor Principal Component Analysis (TPCA) for estimating factors and loadings in a tensor factor model.
result Simple TPCA is optimal for strong factors and can be improved for weak factors with alternating least-squares iterations.
Unified approach to tensor PCA and related problems using tensor cumulants.
problem Statistical inference on invariant distributions, particularly tensor PCA.
method Definition and analysis of tensor cumulants to unify and extend previous results.
result Unified explanation of hardness and subexponential-time algorithms for tensor PCA.
Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Princ…
Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimizati…
A new robust PCA estimator combining M-estimators and minimum divergence estimators.
problem Adverse effect of outlying observations in PCA for high-dimensional data.
method Minimum density power divergence estimator combined with a computationally efficient algorithm.
result High breakdown guarantee regardless of data dimension with theoretical support and practical applications.
RieCUR improves Robust PCA by combining Riemannian optimization and CUR decompositions.
problem Robust Principal Component Analysis (PCA) to recover low-rank and sparse matrices from their sum.
method Riemannian CUR (RieCUR) algorithm that combines Riemannian optimization and robust CUR decompositions.
result RieCUR achieves state-of-the-art performance in Robust PCA with improved robustness to outliers and comparable computational complexity.
Anchor PCA improves robustness in multi-domain PCA.
problem PCA on pooled data can focus on spurious directions.
method Anchor PCA focuses on shared directions of variation.
result Anchor PCA outperforms pooling and worst-case alternatives.
Proposes AE for robust PCA, improving robustness to outliers.
problem PCA's sensitivity to outliers.
method Angular Embedding (AE) and Truncated Angular Embedding (TAE).
result AE/TAE outperforms state-of-the-art RPCA methods.
New algorithm solves fair PCA, robust PCA, and sparse PCA problems efficiently.
problem Fair Principal Component Analysis (FPCA) to ensure fairness in PCA solutions.
method Iterative MM algorithm with SDP reformulation to quadratic program.
result Algorithm monotonically improves fairness objectives at each iteration.
Develops a new tensor PCA method for analyzing multiple network data.
problem Analyzing multiple large networks for dimensionality reduction.
method Semi-Symmetric Tensor PCA (SS-TPCA) for principal components analysis.
result SS-TPCA achieves the same estimation accuracy as classical matrix PCA, with error proportional to the square root of the number of vertices.
It is well known that Principal Component Analysis (PCA) is strongly affected by outliers and a lot of effort has been put into robustification of PCA. In this paper we present a new algorithm for robust PCA minimizing the trimmed reconstruction error. By directly minimizing over the Stiefel manifold, we avoid deflatio…
Proposes MPCA for robust PCA using mode estimation.
problem Outliers sensitivity in PCA.
method Modal Principal Component Analysis (MPCA) based on mode estimation.
result MPCA shows advantages over conventional methods.
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …
Unified PCA framework on flag manifolds for robust data analysis.
problem Outliers and manifold data in PCA.
method Generalization of PCA to flag manifolds, optimization problems, and tangent-PCA integration.
result Novel robust and dual geodesic PCA variations.
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimi…
Robust PCA detects anomalies and fills gaps in seasonal time series data.
problem Anomaly detection and data imputation in seasonal time series.
method Online robust PCA framework for temporal observations.
result Empirically compared and showed effectiveness in practical situations.
Robust PCA, the problem of PCA in the presence of outliers has been extensively investigated in the last few years. Here we focus on Robust PCA in the column sparse outlier model. The existing methods for column sparse outlier model assumes either the knowledge of the dimension of the lower dimensional subspace or the …
SGD recovers multiple signal vectors in noisy tensor PCA.
problem Estimating multiple signal vectors from noisy tensor observations.
method Online stochastic gradient descent (SGD) in high dimensions with detailed analysis of correlations.
result Sequential elimination of correlations allows recovery of all spikes from Np−2 samples. For the tensor PCA (principal component analysis) problem, we propose a new hierarchy of increasingly powerful algorithms with increasing runtime. Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (ap…
Revisits PCA with new formulations and insights.
problem Improving PCA formulations and understanding.
method Difference-of-convex (DC) framework, kernelizability, out-of-sample applicability, simultaneous iteration, DCA perspective.
result PCA-like problems are kernelizable and have new optimization perspectives.
New PCA method handles multiple datasets and detects sparse patterns robustly.
problem Handling multi-source data with sparse and outlier-robust PCA.
method Developed a regularization problem with a penalty for structured sparsity and outlier resistance.
result The method detects global and local patterns across multiple data sources robustly.
Tuning-free OR-PCA improves scalability for large datasets.
problem Dataset sensitivity of OR-PCA tuning parameters.
method Implicit regularization through modified gradient descents.
result Comparable or better performance on simulated and real-world datasets.
A new robust PCA method uses Innovation Search and Leverage Scores.
problem Outlier detection and robust PCA in data clustering.
method Innovation Search and Leverage Scores.
result The method provides theoretical guarantees and outperforms existing algorithms.
SMPI recovers tensor spikes from noisy data with improved performance.
problem Recovering tensor spikes corrupted by Gaussian noise.
method Selective Multiple Power Iterations (SMPI) with polynomial random initializations and symmetrized tensor power iterations.
result SMPI outperforms existing algorithms and approaches theoretical optimal recovery.
New algorithms improve tensor CP decomposition under mild conditions.
problem Improving tensor CP decomposition with theoretical guarantees under mild incoherence conditions.
method Composite PCA and Concurrent Orthogonalization algorithms.
result Theoretical guarantees and practical superiority over existing methods.
Paper proposes an optimal framework for tensor estimation across various applications.
problem Generalized tensor estimation problems in computational imaging, genomics, and network analysis.
method Unified projected gradient descent approach to find low-rank tensor fits under generalized parametric models.
result Achieves minimax optimal rate of convergence in estimation error for various tensor estimation problems.
Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere
problem Estimating rank-R symmetric signal tensor from Gaussian observation
method Profile maximum likelihood estimator
result Finite-(k,d) error bound recovers asymptotically optimal rate