arXiv subjects
Publications and source records attributed to Hari Sarang Nathan.
The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erdős/Falconer type problems.
We introduce a generalized Fourier ratio, the \(\ell^1/\ell^2\) norm ratio of coefficients in an \emph{arbitrary} orthonormal system, as a single, basis-invariant measure of \emph{effective dimension} that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via \(\ell^1\) minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp \emph{localization obstruction}: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.
One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.
Approval voting is a common method of preference aggregation where voters vote by ``approving'' of a subset of candidates and the winner(s) are those who are approved of by the largest number of voters. In approval voting, the degree to which a vote impacts a candidate's score depends only on if that voter approved of the candidate or not, i.e., it is independent of which, or how many, other candidates they approved of. Recently, there has been interest in satisfaction approval voting and quadratic voting both of which include a trade-off between approving of more candidates and how much support each selected candidate gets. Approval voting, satisfaction approval voting, and quadratic voting, can all be viewed as voting where a vote is viewed as analogous to a vector with a different unit norm ($\mathcal{L}^{\infty}$, $\mathcal{L}^{1}$, and $\mathcal{L}^2$ respectively). This suggests a generalization where one can view a vote as analogous to a normalized unit vector under an arbitrary $\mathcal{L}^p$-norm. In this paper, we look at various general methods for justifying voting methods and investigate the degree to which these serve as justifications for these generalizations of approval voting.