CUR decomposition aids in clustering data from unknown subspaces.
problem Clustering data from multiple unknown subspaces.
method CUR decomposition to construct similarity matrices.
result CUR decomposition provides exact clustering in noise-free cases and flexible matrices for noisy data.
A new method for subspace decomposition using an over-complete dictionary.
problem Signal subspace decomposition over dependent basis sets.
method Sparse Signal Subspace Decomposition (3SD) method based on an over-complete dictionary and novel criterion.
result Demonstrates high performance in image denoising.
Online tensor subspace tracking algorithm for incomplete data.
problem Online subspace tracking of partially observed high-dimensional data.
method OLSTEC algorithm based on CP decomposition and recursive least squares.
result OLSTEC outperforms state-of-the-art algorithms in convergence rate.
This paper covers robust subspace learning and tracking methods.
problem Learning and tracking subspaces in the presence of outliers.
method Robust PCA, Robust Subspace Tracking, Robust Subspace Recovery.
result Effective methods for handling outliers in subspace learning and tracking.
Novel NGCA algorithm finds non-Gaussian subspace without iterative steps.
problem Identifying a linear subspace with non-Gaussian projected data.
method Log-density gradient estimation for eigenvalue decomposition.
result Identified subspace converges to true subspace at optimal rate.
Adaptive tensor modeling preserves continuity in multidimensional data.
problem Discretization of continuous multidimensional data loses important information.
method Functional Tucker decomposition (FTD) with RKHS modeling.
result FTD enables adaptive and expressive tensor modeling.
Paper develops a method to identify feature subspaces contributing to local data complexity.
problem Identifying feature subspaces that contribute to local data complexity.
method Develops an estimator of Local Intrinsic Dimension (LID) along axis projections to identify feature subspaces.
result Preliminary evidence suggests LID decomposition can indicate axis-aligned data subspaces supporting cluster formation.
Flow Matching models help generative models stay within the subspace of real data.
problem How do generative models stay within the subspace of real data?
method Flow Matching models using a learned velocity field to transform a simple prior into a complex target distribution.
result Generated samples memorize real data points and represent the sample data subspace exactly.
In multi-label learning, each sample is associated with several labels. Existing works indicate that exploring correlations between labels improve the prediction performance. However, embedding the label correlations into the training process significantly increases the problem size. Moreover, the mapping of the label …
Paper unifies subspace identification and DMD for dynamical systems.
problem Estimating dynamical models from data.
method Unified optimization and regression problems for SID and DMD.
result Proves equivalence of SID and DMD for optimal model construction.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
New method guarantees simultaneous decomposition of tensor components.
problem Existing methods fail to recover all tensor components simultaneously.
method S-ASI method using slicing initialization and subspace iterations.
result Guaranteed recovery of top r components simultaneously for symmetric tensors.
New algorithms extract Koopman invariant subspaces from large-scale data.
problem Difficulty in discerning the Koopman invariant subspace from many Koopman eigenmodes.
method Multi-task feature learning and pruning procedure to remove spurious modes.
result Effective in approximating Koopman operator for complex flows.
In this letter, we consider two sets of observations defined as subspace signals embedded in noise and we wish to analyze the distance between these two subspaces. The latter entails evaluating the angles between the subspaces, an issue reminiscent of the well-known Procrustes problem. A Bayesian approach is investigat…
New findings on GRW space-times with constant scalar curvature.
problem Understanding GRW space-times in different subspaces.
method Analyzing orthogonal subspaces of Gray's decomposition.
result Generalized quasi-Einstein GRW space-times reduce to known types of space-times.
New faster, space-saving methods for subspace embeddings in tensors.
problem Efficiently embedding large tensors with fewer random bits.
method Modewise Johnson-Lindenstrauss embeddings for rank-r tensors. result Improved space complexity for tensor subspaces with fewer random bits.
Paper proposes a fully data-driven method for Koopman spectral analysis.
problem Manual preparation of nonlinear observables is often required for Koopman spectral analysis.
method Learning Koopman invariant subspaces from observed data using linear least-squares regression.
result Performance evaluated using nonlinear dynamical systems and applications.
Study Sp(n)-orbits in complex and Σ-complex subspaces of Hermitian quaternionic vector spaces.
problem Characterize Sp(n)-orbits in Grassmannians of complex and Σ-complex subspaces. method Decompose subspaces into 4-dimensional complex addends and 2-dimensional totally complex subspace. Use properties of isoclinic subspaces and principal angles.
result Determine full set of invariants for Sp(n)-orbits in GrR(2k,4n). The paper studies posets from decompositions in symmetric monoidal categories.
problem Understanding posets from decompositions in symmetric monoidal categories.
method Defining decompositions and partial decompositions, complexes of frames, partial bases, and ordered versions.
result Unified approach to combinatorics and homotopy type of posets and complexes.
Neural-ANOVA breaks down neural networks into simpler models.
problem Understanding complex neural network decision-making processes.
method Formulates a learning problem to decompose neural networks into lower-order models using ANOVA.
result Demonstrates improved approximation properties compared to other regression methods.
The paper proposes a method to balance fairness and prediction accuracy by adjusting data representations.
problem Machine learning models can inherit and amplify historical biases, leading to unfair outcomes.
method The paper uses subspace decomposition and influence analysis to control the fairness-utility trade-off.
result The method effectively improves fairness while preserving predictive performance.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
This work improves smoothed analysis for several unsupervised learning problems.
problem Overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis.
method Developed high-confidence lower bounds on the least singular value of structured random matrix ensembles and used them to design algorithms with polynomial time smoothed analysis guarantees.
result Polynomial time smoothed analysis guarantees for robust subspace recovery, learning overcomplete hidden markov models, and higher order tensor decompositions.
GEOMANCER learns manifold factors without supervision.
problem Learning to factorize Lie group orbits from data.
method Subspace diffusion estimation of invariant subspaces.
result Unsupervised disentanglement of complex manifolds possible.
This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a covariant functor from the category of finite-dimensional diffeological vector …
Novel method decomposes configuration space for improved collision checking.
problem Improving collision checking in high-degree-of-freedom robot motion planning.
method Proposes a configuration space decomposition method to build a composite classifier.
result Composite classifier outperforms state-of-the-art single classifier methods.
This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which…
GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an incremental algorithm for identifying a subspace of Rn from a sequence of vectors in this subspace, where only a subset of components of each vector is revealed at each iteration. Recent analysis has shown that GROUSE converges locally at an expected linea…
Paper detects adversarial attacks in sound classification models.
problem Adversarial attacks threaten data-driven models, especially in sound classification.
method Detects adversarial subspaces in unitary vector domain using chordal distance and generalized Schur decomposition.
result Regularized logistic regression detector outperforms other approaches on benchmark datasets.
Proposes a method to predict responses from covariates over time.
problem Predicting responses from covariates with changing conditional distributions over time.
method Invariant Subspace Decomposition (ISD) framework that splits the conditional distribution into time-invariant and time-dependent components.
result The decomposition can be used for zero-shot and time-adaptation prediction tasks.
New method disentangles hidden data structures using HSIC and supervision.
problem Tackles the challenge of interpreting high-dimensional data.
method Supervised Independent Subspace Principal Component Analysis (sisPCA) using HSIC.
result Identifies and separates hidden data structures effectively.
Sparse spectral decomposition identifies overlapping communities in networks.
problem Estimating overlapping community memberships in networks where nodes can belong to multiple communities.
method Sparse principal subspace estimation with iterative thresholding.
result The fixed point of the algorithm corresponds to correct node memberships under the stochastic block model.
A new model reduces noise and speeds up subspace segmentation.
problem Subspace segmentation from noisy data.
method Group norm regularized factorization model (GNRFM) with AALM algorithm.
result The method is faster and more robust to noise.
New Krylov subspace methods speed up mixed-effects models with crossed random effects.
problem Slow computations for high-dimensional crossed random effects in mixed-effects models.
method Krylov subspace-based methods for generalized mixed-effects models with cross effects.
result Speedups by factors of up to 10,000 in computations for mixed-effects models.
Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W^{\bot} of W within the dual space of K[S_r] yield linear identities needed for a t…
New method accelerates neural network training by focusing on flat directions.
problem Improving neural network training speed and stability.
method Bulk-SGD, interpolated gradient methods.
result Updates along the Dominant subspace can accelerate convergence but compromise stability.
Faster matrix completion through randomized SVD algorithms.
problem Efficiently completing large sparse matrices for applications like image inpainting and recommender systems.
method Proposed two fast randomized algorithms (rSVD-PI and rSVD-BKI) and a new subspace recycling technique to accelerate singular value thresholding (SVT) method.
result The proposed algorithms achieve up to 15X faster computation time for image inpainting and movie rating estimation problems.
Tyler's M-estimator's phase transition at DS-SNR = 1 is resolved.
problem Robust Subspace Recovery
method Tyler's M-estimator
result TME converges exactly to the true subspace for DS-SNR >= 1 under a new stability condition.
Randomly shuffled kernels can be compressed efficiently.
problem Reducing storage cost of CNN parameters on resource-limited platforms.
method Randomly-shuffled tensor decomposition (RsTD) to embed kernels into random low-rank subspaces.
result CNNs can be significantly compressed even with randomly shuffled kernels, achieving more stable accuracy.
Efficiently compress pretrained models using RSI for improved predictive accuracy.
problem Efficiently compressing large pretrained models for practical deployment.
method Randomized subspace iteration (RSI) for low-rank approximation of pretrained models.
result RSI achieves near-optimal approximation quality and outperforms RSVD in predictive accuracy.
The paper updates SVD of evolving matrices using projection techniques.
problem Updating the rank-k truncated SVD of evolving matrices.
method Projection viewpoint, building subspaces to approximate singular vectors.
result The proposed algorithm leads to higher accuracy, especially for large singular values.
Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency mat…
This work improves tensor decomposition methods, especially for large datasets.
problem Lack of efficient methods for estimating Tucker decompositions.
method Applies Johnson-Lindenstrauss type guarantees to Tucker decompositions with random embeddings.
result Effective dimension reduction with minimal error for large tensors.
BankGCN improves graph convolution networks by handling multi-channel signals with adaptive filter banks.
problem Handling multi-channel graph signals with limited architectures.
method BankGCN decomposes multi-channel signals into subspaces and uses adapted filters for each subspace.
result BankGCN achieves excellent performance in graph classification on benchmark datasets.
The study examines algebraic structures of specific tensor forms in four-dimensional spacetimes.
problem Investigating algebraic features of certain tensor forms in spacetimes.
method General treatment followed by specialization to four-dimensional spacetimes, focusing on invariant subspaces and generalizing relations.
result Generalized relations such as the Ruse-Lanczos identity, Bel-Matte decomposition, and Lovelock-like quadratic identities.
New conditions for GRW space-times to be perfect-fluid space-times.
problem Conditions for GRW space-times to be perfect-fluid.
method Gray's decomposition of the gradient of the Ricci tensor, determining Ricci tensor forms in invariant subspaces.
result For most GRW space-times, the Ricci tensor is Einstein or perfect fluid.
Study optimizes shared singular subspace estimation from noisy matrices.
problem Estimating shared singular subspaces across multiple noisy matrices.
method Low-rank matrix denoising framework with Stack-SVD and novel estimators.
result Stack-SVD achieves minimax rate-optimality for identical shared subspaces, and novel estimators for partial sharing.
The paper finds Koopman invariant subspaces using personalized PageRank.
problem Selecting a finite dictionary of observables for Koopman-invariant span.
method Exploiting zero-block structure in EDMD matrices and applying PageRank.
result Personalized PageRank can detect Koopman invariant subspaces.