Related papers: Hardness of High-Dimensional Linear Classification
We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely…
We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ and sampling oracle access to an unknown distribution…
To model modern large-scale datasets, we need efficient algorithms to infer a set of $P$ unknown model parameters from $N$ noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise…
Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree…
High-dimensional statistical inference with general estimating equations are challenging and remain less explored. In this paper, we study two problems in the area: confidence set estimation for multiple components of the model parameters,…
We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
Superposition, the ability of neural networks to represent more features than neurons, is increasingly seen as key to the efficiency of large models. This paper investigates the theoretical foundations of computing in superposition,…
In [R. Andreani, G. Haeser, L. M. Mito, H. Ram\'irez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson's constraint…
Allen's interval algebra is one of the most well-known calculi in qualitative temporal reasoning with numerous applications in artificial intelligence. Recently, there has been a surge of improvements in the fine-grained complexity of…
We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…
We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation…
We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its…
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in…
A fundamental challenge in semi-supervised learning lies in the observed data's disproportional size when compared with the size of the data collected with missing outcomes. An implicit understanding is that the dataset with missing…
In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated…
In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb R^n$ , $1 \leq d \leq n-1$. We show that a canonical affine invariant measure exists and…
Label distribution (LD) uses the description degree to describe instances, which provides more fine-grained supervision information when learning with label ambiguity. Nevertheless, LD is unavailable in many real-world applications. To…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
High-dimensional classification is a fundamentally important research problem in high-dimensional data analysis. In this paper, we derive a nonasymptotic rate for the minimax excess misclassification risk when feature dimension…