arXiv research

The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.

In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line (respectively an inter…

Adam achieves optimal convergence in deep ReLU networks via novel Kakeya bounds.

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all…

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

Proves lower bounds on Hausdorff dimension of projections of invariant sets.

We use Toponogov's triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya-Nikodym norms introduced in \cite{SKN} for manifolds of nonpositive sectional curvature. Using thes…

We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee …

Improved spectral projection estimates on manifolds of non-positive curvature.

New lower bounds for sampling from log-concave distributions in higher dimensions.

Michael-Simon inequality proven for anisotropic energies close to area.

We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms ove…

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

The geometry of oscillatory integrals on manifolds with intermediate symmetry.

Study shows Julia sets and gasket limit sets are quasiconformally different.

Study shows non-symmetric convex sets have full boundary limits.

The paper analyzes set-to-set matching with neural networks, focusing on theoretical generalization.

Generative model learns to autoencode and generate sets of images.

Study on cold and freezing sets in digital images.

Paper solves whether zero sets are mapping degree sets.

Matching two different sets of items, called heterogeneous set-to-set matching problem, has recently received attention as a promising problem. The difficulties are to extract features to match a correct pair of different sets and also preserve two types of exchangeability required for set-to-set matching: the pair of …

New set-valued star-shaped risk measures introduced for better risk assessment.

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…

Study dynamics and topology of flows near non-saddle sets or W-sets.

The study explores mapping degree sets and their properties for manifolds.

Current approaches for predicting sets from feature vectors ignore the unordered nature of sets and suffer from discontinuity issues as a result. We propose a general model for predicting sets that properly respects the structure of sets and avoids this problem. With a single feature vector as input, we show that our m…

This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.

Consider a general machine learning setting where the output is a set of labels or sequences. This output set is unordered and its size varies with the input. Whereas multi-label classification methods seem a natural first resort, they are not readily applicable to set-valued outputs because of the growth rate of the o…

Deep Sets approximates functions on sets with high-dimensional latent space.

Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.

A stability-based method selects the most desirable conformal prediction set.

The paper links set cuspidality to function regularity and flatness.

This work establishes properties on diffeological structures for set-valued maps and measures.

Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.

This letter introduces an abstract learning problem called the "set embedding": The objective is to map sets into probability distributions so as to lose less information. We relate set union and intersection operations with corresponding interpolations of probability distributions. We also demonstrate a preliminary so…

Find limiting sets for digital cones and suspensions.

Analytic sets with unique infinite tangent cone are algebraic.

Study freezing sets for digital images in a 2D grid.

Fuzzy prediction sets generalize binary predictions to include elements at varying confidence levels.

Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…

Develops deep neural network techniques for sets as input and output.

Proves a theorem for Assouad dimension with applications to distance sets and radial projections.

The paper explores connections between perimeter, area, and visual angle of convex sets.

The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.

New tools for constructing fixed point sets in digital topology.

Set risk measures extend traditional risk measures to handle sets of positions.

Study contractibility of boundaries in convex sets and limit sets of subgroups.

Set classification problems arise when classification tasks are based on sets of observations as opposed to individual observations. In set classification, a classification rule is trained with $N$ sets of observations, where each set is labeled with class information, and the prediction of a class label is performed a…