CAVI converges exponentially fast for Bayesian PCA models.
problem Characterizing the convergence speed of CAVI for BPCA.
method Proved exponential convergence using power iteration analogy and novel lower bounds.
result Exponential convergence of CAVI for BPCA models with any number of principal components.
Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner. However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since ea…
Principal component analysis (PCA) is a popular method for projecting data onto uncorrelated components in lower dimension, although the optimal number of components is not specified. Likewise, multiple signal classification (MUSIC) algorithm is a popular PCA-based method for estimating directions of arrival (DOAs) of …
EB-PCA reduces noise in high-dimensional PCA by estimating a joint prior distribution.
problem High-dimensional PCA noise in samples comparable to or larger than data.
method Empirical Bayes PCA using Kiefer-Wolfowitz MLE, random matrix theory, and AMP algorithm.
result EB-PCA achieves Bayes-optimal accuracy in spiked models and significantly improves over PCA in simulations and real data.
Improves inverse uncertainty quantification for time-dependent data using PCA and deep neural networks.
problem Efficiently quantify model input uncertainties from time-dependent experimental data.
method Functional PCA for dimensionality reduction, deep neural networks for surrogate modeling, Bayesian neural networks for uncertainty estimation.
result The proposed method reduces the computational cost and improves the agreement with experimental data.
Probabilistic Autoencoder learns latent space weights' distribution.
problem Nonlinear model reconstruction error and sample quality.
method Normalizing flow for latent space weights' probability distribution.
result PAE achieves small reconstruction errors, high sample quality, and good performance.
Commonly used in computer vision and other applications, robust PCA represents an algorithmic attempt to reduce the sensitivity of classical PCA to outliers. The basic idea is to learn a decomposition of some data matrix of interest into low rank and sparse components, the latter representing unwanted outliers. Althoug…
We consider probabilistic PCA and related factor models from a Bayesian perspective. These models are in general not identifiable as the likelihood has a rotational symmetry. This gives rise to complicated posterior distributions with continuous subspaces of equal density and thus hinders efficiency of inference as wel…
KPCA-BO improves BO for high-dimensional optimization problems by learning a non-linear sub-manifold.
problem High-dimensional optimization problems where Gaussian Process regression requires too much data and computation.
method KPCA-BO embeds a non-linear sub-manifold in the search space, learning a GPR model on this sub-manifold.
result KPCA-BO outperforms vanilla BO in convergence speed, especially as dimensionality increases.
Novel method converts time series data into functional data for high dimensional classification.
problem Small sample size problem in high dimensional time series data.
method Classwise Functional Principal Component Analysis (PCA) followed by Bayesian linear classifier.
result Demonstrated efficacy on synthetic and real data sets.
We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal-gamma prior distribution. In this context, we exhibit a closed-form expression o…
Principal component analysis (PCA) is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Only few works have proposed a probabilistic approach able to infer the number of significant c…
Bayesian method reconstructs neural network memories from connectivity.
problem Reconstructing memories from neural network connectivity.
method Bayesian inference using statistical physics principles.
result Algorithm successfully reconstructs stored patterns from synaptic connectivity.
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…
Simplifies fair PCA with fast, efficient solution.
problem Learning fair low-rank approximations of data.
method Conceptually simple approach with analytic solution.
result Faster and similar results to existing fair PCA methods.
KAN-PCA improves asset return analysis by capturing more variance than classical PCA during market crises.
problem Inefficient classical PCA during market crises when correlations between assets change dramatically.
method KAN-PCA uses KAN (Kolmogorov-Arnold Networks) with B-spline functions to learn nonlinear projections.
result KAN-PCA achieves a higher reconstruction R^2 (66.57%) compared to classical PCA (62.99%) on 20 S&P 500 stocks.
Deep learning is a form of machine learning for nonlinear high dimensional pattern matching and prediction. By taking a Bayesian probabilistic perspective, we provide a number of insights into more efficient algorithms for optimisation and hyper-parameter tuning. Traditional high-dimensional data reduction techniques, …
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.
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.
TL-PCA uses transfer learning to improve PCA performance with limited target data.
problem PCA performance is limited with scarce target data.
method Transfer learning approach to PCA (TL-PCA) that combines source task knowledge with target task data.
result Improved PCA representation for dimensionality reduction with limited target data.
Bucketed PCA-NN outperforms DNNs by 96% on MNIST.
problem Benchmarking deep neural networks for supervised classification.
method Applies PCA to individual buckets constructed in two phases, retains neural network architecture, and uses neurons that mirror input signals.
result Bucketed PCA-NN achieves 96% accuracy on MNIST, similar to DNNs.
A new low-dimensional parameterization based on principal component analysis (PCA) and convolutional neural networks (CNN) is developed to represent complex geological models. The CNN-PCA method is inspired by recent developments in computer vision using deep learning. CNN-PCA can be viewed as a generalization of an ex…
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.
A new method for fair PCA ensures balanced error across groups.
problem Balancing approximation error across different groups in multi-group data.
method Iterative method to compute fair principal components minimizing max group-wise reconstruction error.
result Preserves the containment property of standard PCA and reduces to standard PCA for single-group data.
This is a detailed tutorial paper which explains the Principal Component Analysis (PCA), Supervised PCA (SPCA), kernel PCA, and kernel SPCA. We start with projection, PCA with eigen-decomposition, PCA with one and multiple projection directions, properties of the projection matrix, reconstruction error minimization, an…
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.
Proposes σ-PCA to learn identifiable linear transformations without whitening.
problem Cannot identify axes with equal variances in PCA.
method Unified model for linear and nonlinear PCA, introducing a missing piece to eliminate rotational indeterminacy.
result Eliminates subspace rotational indeterminacy in PCA.
Solution to sparse PCA tuning problem using Empirical Bayes.
problem Sparse PCA multiple tuning problem (MTP).
method Empirical Bayes covariance decomposition for penalized PCA.
result Empirical Bayes approach efficiently solves MTP in sparse PCA.
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.
New analysis improves black-box k-PCA algorithms, reducing parameter loss.
problem Designing efficient k-PCA algorithms with black-box access to a 1-PCA oracle. method Black-box deflation methods, analyzing ePCA and cPCA approximations.
result Deflation methods suffer no asymptotic parameter loss for k-cPCA in feasible regimes. Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
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.
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…
A new PCA method using Tℓ1-norm outperforms existing methods.
problem Outliers and noise sensitivity in classical PCA.
method PCA based on Tℓ1-norm maximization. result The method outperforms PCA-ℓp, ℓpSPCA, and PCA in numerical experiments. New fair PCA method using streaming algorithms with statistical guarantees.
problem Perform PCA while ensuring fair projected distributions.
method Formulated new notion PAFO learnability, proposed fair noisy power method (FNPM) for memory efficiency.
result First statistical guarantee for fair PCA in streaming setting.
PCA-Guided Quantile Sampling preserves data structure in large datasets.
problem Preserving data structure in large-scale subsampling.
method PCA QS uses leading principal components to guide stratified sampling.
result PCA QS maintains statistical and geometric structure without distorting semantics.
A method to remove mean-shift noise from PCA using knockoffs.
problem High sensitivity of PCA to mean-shift contamination in high-dimensional data.
method Introducing knockoff mean-shift perturbation to separate and remove mean-shift components from PCA.
result The mean-shift spikes are spectrally separable from stable eigenvalues, allowing for robust PCA.
AgFlow speeds up model selection in penalized PCA.
problem Efficient model selection in penalized PCA for HDLSS settings.
method Implicit regularization effect of gradient flow to reduce computation complexity.
result AgFlow achieves the complete solution path of L2-penalized PCA.
One technique to visualize the training of neural networks is to perform PCA on the parameters over the course of training and to project to the subspace spanned by the first few PCA components. In this paper we compare this technique to the PCA of a high dimensional random walk. We compute the eigenvalues and eigenvec…
GT-PCA improves PCA for image and time series data.
problem Lack of robustness to transformations in PCA.
method GT-PCA is a neural network that estimates components invariant to specific transformations.
result GT-PCA outperforms alternative methods in synthetic and real data experiments.
Modified PCA algorithm with continual learning preserves features of previous modes for multimode process monitoring.
problem Catastrophic forgetting of previous modes in monitoring models for successive modes.
method Modified PCA algorithm with elastic weight consolidation (EWC) to preserve features of previous modes.
result PCA-EWC algorithm effectively monitors multimode processes without performance decrease.
PCA-Net combines PCA and neural networks for operator approximation, with new bounds on complexity.
problem Developing approximation theory for PCA-Net architecture.
method Combines PCA and neural networks, derives universal approximation results and lower bounds on complexity.
result PCA-Net can overcome the curse of parametric complexity for specific operators.
A new dynamical formulation of log-PCA captures local principal modes of geodesic variations.
problem Learning principal variations of random probability measures under Wasserstein geometry.
method Introducing a new dynamical formulation of log-PCA as a variational approach.
result Deriving a general statistical convergence rate for empirical WT-PCA.
We performed an empirical comparison of ICA and PCA algorithms by applying them on two simulated noisy time series with varying distribution parameters and level of noise. In general, ICA shows better results than PCA because it takes into account higher moments of data distribution. On the other hand, PCA remains quit…
Generative Kernel PCA explores latent spaces for data interpretation and novelty detection.
problem Exploring latent spaces of datasets for better data interpretation.
method Generative Kernel PCA using hidden and visible units similar to Restricted Boltzmann Machines.
result Gradually moving in the latent space allows for interpretation of components and detection of novel patterns.
Principal Component Analysis (PCA) has been used to study the pathogenesis of diseases. To enhance the interpretability of classical PCA, various improved PCA methods have been proposed to date. Among these, a typical method is the so-called sparse PCA, which focuses on seeking sparse loadings. However, the performance…
A fair PCA method using JEVD ensures balanced data representation.
problem PCA's bias in data with demographic characteristics.
method Joint Eigenvalue Decomposition (JEVD) for fair PCA.
result JEVD optimally balances fairness and PCA's data structure.
Two new PCA variants improve financial data analysis.
problem Numerical instability and nonstationarity in PCA for finance.
method Iterated and exponentially weighted moving PCA variants using Ogita-Aishima iteration.
result Improved stability and adaptability in financial data analysis.