Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
Optimal subspace embedding with near-optimal sparsity for high-dimensional data.
problem Efficiently preserving norms of vectors in high-dimensional subspaces.
method Near-optimal sparsity oblivious subspace embedding with decoupling argument and cumulant method.
result Achieved near-optimal sparsity of O~(1/ε) non-zeros per column. Proposes a diagnostic method to evaluate factor models using cap-axis integrals.
problem Improving factor model evaluation in low-dimensional spaces.
method Lifts pricing errors into a bridge-alpha curve along the market-capitalization rank axis.
result The cap-axis norm is distinct from Sharpe gain and size exposure.
Proposes a diagnostic method to evaluate factor models using cap-axis integrals.
problem Improving factor model evaluation for low-dimensional models.
method Lifts pricing errors into a bridge-alpha curve along the market-capitalization rank axis.
result The cap-axis norm is distinct from Sharpe gain and size exposure.
When data is sampled from an unknown subspace, principal component analysis (PCA) provides an effective way to estimate the subspace and hence reduce the dimension of the data. At the heart of PCA is the Eckart-Young-Mirsky theorem, which characterizes the best rank k approximation of a matrix. In this paper, we prove …
State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with ℓ1, ℓ2 or nuclear norms. ℓ1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no conne…
Study on tensor nuclear norm's decomposability and subdifferential.
problem Understanding tensor nuclear norm in higher-order tensors.
method Showed decomposability over specific subspaces, derived subdifferential inclusions, and studied subgradients.
result Established the statistical performance of tensor robust principal component analysis.
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 method improves subspace iteration for eigenvectors in machine learning.
problem Computing eigenvectors for large-scale problems in machine learning.
method Subspace iteration with ℓ2o∞ norm convergence analysis. result Deterministic bounds and practical stopping criterion for improved performance.
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
New research shows SSC fails when points on the same subspace are mislabeled.
problem Failure of SSC when points on the same subspace are mislabeled.
method Analyzed the effect of different distributions of points on the same subspace.
result SSC fails to infer correct labels when points on the same subspace fall into more than one cluster.
Proposes new ℓ0-based methods for low-rank sparse subspace clustering.
problem Clustering high-dimensional data points represented by low-dimensional subspaces.
method Introduces two ℓ0 quasi-norm based regularizations: GMC-LRSSC and S0/ℓ0-LRSSC. Solves resulting nonconvex optimization problems using alternating direction method of multipliers. result Demonstrates effectiveness of proposed methods on synthetic and real-world datasets.
Proposes an optimization framework for sparse robust subspace estimation.
problem Sparse robust one-dimensional subspace estimation.
method l1-norm regularization, linear relaxation, simple ratios, sorting techniques.
result Achieves global optimality for sparse robust subspace with polynomial time efficiency.
We develop embeddings for nonlinear subspaces preserving vector norms.
problem Preserving vector norms in nonlinear subspaces.
method Low-distortion embeddings for subspaces under nonlinear transformations.
result First low-distortion embeddings for a wide class of nonlinear functions.
The study analyzes perturbation bounds for HOSVD and introduces new tensor denoising estimators.
problem Perturbation analysis of HOSVD under random noise.
method Developed sup-norm perturbation bounds and introduced new tensor denoising estimators.
result Sharp deviation bounds in the sup-norm for singular subspaces and fast convergence rate for tensor denoising.
We consider the problem of recovering a low-rank tensor from its noisy observation. Previous work has shown a recovery guarantee with signal to noise ratio O(n⌈K/2⌉/2) for recovering a Kth order rank one tensor of size n×⋯×n by recursive unfolding. In this paper, we first improve…
Subspace clustering methods based on ℓ1, ℓ2 or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, ℓ1 regularization is guaranteed t…
We describe ways to define and calculate L1-norm signal subspaces which are less sensitive to outlying data than L2-calculated subspaces. We focus on the computation of the L1 maximum-projection principal component of a data matrix containing N signal samples of dimension D and conclude that the general proble…
Paper proposes equivalent Lipschitz surrogates for zero-norm and rank optimization problems.
problem Optimization problems involving zero-norm and rank functions.
method Reformulate as MPECs, use global exact penalty, eliminate dual variable to get surrogates.
result Obtained equivalent Lipschitz surrogates for zero-norm and rank optimization problems.
Matrix rank minimizing subject to affine constraints arises in many application areas, ranging from signal processing to machine learning. Nuclear norm is a convex relaxation for this problem which can recover the rank exactly under some restricted and theoretically interesting conditions. However, for many real-world …
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering ℓ∞ and ℓ2,∞ bounds. result Fine-grained insights into singular vector and subspace perturbations, including ℓ∞ and ℓ2,∞ bounds. Paper analyzes convergence of PAM method for low-rank factorization models.
problem Convergence analysis of PAM method with subspace correction for low-rank factorization models.
method Majorized proximal alternating minimization (PAM) method with subspace correction.
result Established full convergence of PAM method under KL property and column ℓ2,0-norm condition. In this paper, we present GASG21 (Grassmannian Adaptive Stochastic Gradient for L2,1 norm minimization), an adaptive stochastic gradient algorithm to robustly recover the low-rank subspace from a large matrix. In the presence of column outliers, we reformulate the batch mode matrix L2,1 norm minimization with…
A new method clusters multi-view data by sharing a common trace-norm of coefficient matrices.
problem Insufficient exploitation of multi-view data due to uniform coefficient matrices.
method Imposes bilinear factorization with orthonormality and low-rank constraints on coefficient matrices.
result The proposed CBF-MSC method effectively clusters multi-view data more comprehensively.
The paper characterizes functions of shallow ReLU NN denoisers under minimal norm constraints.
problem Understanding the theoretical success of neural network denoisers.
method Characterization of functions realized by shallow ReLU NN denoisers under minimal norm constraints.
result The functions realized by shallow ReLU NN denoisers are contractive toward clean data points and generalize better than the empirical MMSE estimator at low noise levels.
GCNs' performance linked to feature, graph, and ground truth alignment.
problem Improving GCNs' classification performance.
method Subspace alignment measure (SAM) based on Frobenius norm of chordal distances.
result SAM quantifies the alignment between features, graph, and ground truth.
Paper introduces S-SSE for stable sparse subspace embedding.
problem Inefficient sparse random projection matrices with uneven non-zero distribution.
method Uses uniform sampling without replacement to create a stable sparse subspace embedded matrix (S-SSE).
result S-SSE maintains Euclidean distance better after dimension reduction.
Study robust estimation of principal components under adversarial perturbations.
problem Estimating principal components in high-dimensional data under adversarial perturbations.
method Design of a computationally efficient algorithm for recovering the top-r principal subspace.
result The algorithm recovers an estimate of the top-r principal subspace with error depending on the robustness parameter κ.
Study on low-dimensional adversarial perturbations in classification models.
problem Understanding and quantifying the effectiveness of low-dimensional adversarial perturbations.
method Analytical lower-bounds for fooling rate, considering binary classifiers under generic regularity conditions.
result Rigorous explanation for the success of heuristic methods in generating low-dimensional adversarial perturbations.
We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in reproduci…
Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. …
GAME improves matrix completion by considering subgroup-specific latent structures.
problem Heterogeneous data with overlapping categories, smoothing away subgroup-specific variation.
method Group-Aware Matrix Estimation (GAME) with overlapping nuclear-norm penalties.
result GAME outperforms global low-rank estimators in structured missingness regimes.
Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
This paper presents GRASTA (Grassmannian Robust Adaptive Subspace Tracking Algorithm), an efficient and robust online algorithm for tracking subspaces from highly incomplete information. The algorithm uses a robust l1-norm cost function in order to estimate and track non-stationary subspaces when the streaming data …
Matrix rank minimization problem is in general NP-hard. The nuclear norm is used to substitute the rank function in many recent studies. Nevertheless, the nuclear norm approximation adds all singular values together and the approximation error may depend heavily on the magnitudes of singular values. This might restrict…
Subspace identification is a classical and very well studied problem in system identification. The problem was recently posed as a convex optimization problem via the nuclear norm relaxation. Inspired by robust PCA, we extend this framework to handle outliers. The proposed framework takes the form of a convex optimizat…
Simply connected spaces of tight frames identified.
problem Understanding the connectivity of spaces of tight frames.
method Viewing tight frames as elements of Stiefel manifolds and identifying simply connected spaces.
result Spaces of tight frames, including finite unit-norm tight frames, are simply connected.
We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having m=O(r~/ε2) rows. Here r~ is the maximum stable rank, i.e. squared ratio of Frobenius and op…
Stochastic Sparse Subspace Clustering improves subspace clustering by reducing over-segmentation through dropout.
problem Over-segmentation in subspace clustering.
method Introducing dropout regularization to enforce denser connections between points from the same subspace.
result Stochastic Sparse Subspace Clustering effectively handles large datasets and reduces over-segmentation.
Paper analyzes SSC for data with missing entries, improving performance.
problem Theoretical analysis of SSC with missing data entries.
method Analyzes theoretical guarantees for SSC with incomplete data, projecting zero-filled data onto observation pattern.
result Improves performance of SSC with incomplete data by projecting zero-filled data onto observation pattern.
The paper proves a unique orbit for a specific genus 3 curve.
problem Proving the uniqueness of a closed orbit in genus 3.
method Understanding the Forni subspace and solving the jump problem.
result The Eierlegende Wollmilchsau orbit is the only one with zero Lyapunov exponent.
The paper explores why a specific type of predictor works well in noisy data.
problem Understanding why a specific type of predictor (minimum-norm interpolator) works well in noisy data.
method The paper uses uniform convergence and zero-error predictors in a norm ball to explain the success of the minimum-norm interpolator.
result The minimum-norm interpolator is consistent, and this can be explained by uniform convergence of zero-error predictors in a norm ball.
Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for {\em real-time} processing of streaming data pose major challenge…
A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the lit…
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.
Different space/time splittings lead to non-equivalent norms on spinor bundles.
problem Non-equivalent norms on spinor bundles due to different space/time splittings.
method Exploration of generalized Doppler shift between maximal negative definite subspaces.
result Necessary and sufficient condition for equivalent norms in terms of Doppler shift.
Paper improves robust subspace clustering for noisy and missing data.
problem Clustering points on multiple subspaces with noise and missing data.
method Robust variant of sparse subspace clustering (SSC) with explicit noise and missing data tolerance bounds.
result Establishes clustering guarantees for higher tolerance to noise and missing data.
A new geometry-preserving method for interpreting compositional data.
problem Statistical challenges in high-dimensional compositional data.
method Geometry-preserving framework for dimension reduction of compositional data.
result Identification of a central compositional subspace for compositional predictors.