arXiv research

Study on minimal hypersurfaces in a special normed space.

The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.

Study on minimal surfaces in a 3D space with 2m-norm.

Hexagonal norm double bubble problem solved with minimal configurations.

We address some theoretical guarantees for Schatten-$p$ quasi-norm minimization ($p \in (0,1]$) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition…

Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.

Formula proves monotonicity for anisotropic minimal hypersurfaces.

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…

Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to recover…

We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $\mathbb{R}^2$.

An elementary proof found for the double bubble problem in a specific norm.

We consider the question of what functions can be captured by ReLU networks with an unbounded number of units (infinite width), but where the overall network Euclidean norm (sum of squares of all weights in the system, except for an unregularized bias term for each unit) is bounded; or equivalently what is the minimal …

In this paper we investigate panel regression models with interactive fixed effects. We propose two new estimation methods that are based on minimizing convex objective functions. The first method minimizes the sum of squared residuals with a nuclear (trace) norm regularization. The second method minimizes the nuclear …

Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we…

The Schatten quasi-norm was introduced to bridge the gap between the trace norm and rank function. However, existing algorithms are too slow or even impractical for large-scale problems. Motivated by the equivalence relation between the trace norm and its bilinear spectral penalty, we define two tractable Schatten norm…

The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function, and is the tighter approximation to matrix rank. However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as well as for nuclear norm minimization, are too slow or even impractical for large-scale problem…

New proof shows norms can't explain deep learning's implicit regularization.

The paper analyzes the performance of empirical risk minimization for $p$-norm linear regression.

As one of the most popular linear subspace learning methods, the Linear Discriminant Analysis (LDA) method has been widely studied in machine learning community and applied to many scientific applications. Traditional LDA minimizes the ratio of squared L2-norms, which is sensitive to outliers. In recent research, many …

The paper constructs a lamination related to minimal hypersurfaces calibrated by a cohomology class.

Rank minimization has attracted a lot of attention due to its robustness in data recovery. To overcome the computational difficulty, rank is often replaced with nuclear norm. For several rank minimization problems, such a replacement has been theoretically proven to be valid, i.e., the solution to nuclear norm minimiza…

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in \cite{tang2012csotg} to recove…

The Hopf fibration is rigid among minimal maps between spheres.

The paper tackles multi-armed bandits with vector losses, focusing on minimizing the $\ell^\infty$-norm of relative losses.

The Schatten-p quasi-norm $(0<p<1)$ is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may…

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions g…

New insights into network generalization show learning rate affects both norm and sharpness.

Optimal financial strategies minimize risk under uncertain models.

The paper explores why a specific type of predictor works well in noisy data.

Let $L$ be an $n$-component link ($n>1$) with pairwise nonzero linking numbers in a rational homology $3$-sphere $Y$. Assume the link complement $X:=Y\setminusν(L)$ has nondegenerate Thurston norm. In this paper, we study when a Thurston norm-minimizing surface $S$ properly embedded in $X$ remains norm-minimizing after…

SAM improves generalization in overparameterized models, but its behavior in tensorized models is less understood.

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that …

Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discont…

Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.

Support Vector Machine (SVM) is an efficient classification approach, which finds a hyperplane to separate data from different classes. This hyperplane is determined by support vectors. In existing SVM formulations, the objective function uses L2 norm or L1 norm on slack variables. The number of support vectors is a me…

This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables an…

Unified algorithm for minimizing composite functions with flexible design.

The study analyzes robustness of estimators in linear models with adversarial errors.

Improved lower bound for geodesics on manifolds.

New method for factor analysis using nuclear and $\ell_0$ norms.

Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the…

ResNets minimize circuit size for fitting data in HTMC regime.

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

New insights into optimization and generalization for linear models.

Many practical applications such as gene expression analysis, multi-task learning, image recognition, signal processing, and medical data analysis pursue a sparse solution for the feature selection purpose and particularly favor the nonzeros \emph{evenly} distributed in different groups. The exclusive sparsity norm has…

New method for efficient proximal mapping of 1-path-norm in shallow networks.

Paper tackles low-rank matrix recovery with column $\ell_{2,0}$-norm regularization.

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.