arXiv subjects

Anand Hemmady

Publications and source records attributed to Anand Hemmady.

3 recordsLinked to original sources

arXiv2026

Local EGOP for Continuous Index Learning

We introduce the setting of continuous index learning, in which a function of many variables varies only along a small number of directions at each point. For efficient estimation, it is beneficial for a learning algorithm to adapt, near each point $x$, to the subspace that captures the local variability of the function $f$. We pose this task as kernel adaptation along a manifold with noise, and introduce Local EGOP learning, a recursive algorithm that utilizes the Expected Gradient Outer Product (EGOP) quadratic form as both a metric and inverse-covariance of our target distribution. We prove that Local EGOP learning adapts to the regularity of the function of interest, showing that under a supervised noisy manifold hypothesis, intrinsic dimensional learning rates are achieved for arbitrarily high-dimensional noise. Empirically, we compare our algorithm to the feature learning capabilities of deep learning. Additionally, we demonstrate improved regression quality compared to two-layer neural networks in the continuous single-index setting.

stat.ML↗

arXiv2018

On algorithms to calculate integer complexity

We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erdős, Guy, Isbell, Selfridge, and others. Let $f(n)$ be the complexity of $n \in \mathbb{Z^{+}}$, where $f(n)$ is defined as the least number of $1$'s needed to represent $n$ in conjunction with an arbitrary number of $+$'s, $*$'s, and parentheses. Several algorithms have been developed to calculate the complexity of all integers up to $n$. Currently, the fastest known algorithm runs in time $\mathcal{O}(n^{1.230175})$ and was given by J. Arias de Reyna and J. van de Lune in 2014. This algorithm makes use of a recursive definition given by Guy and iterates through products, $f(d) + f\left(\frac{n}{d}\right)$, for $d \ |\ n$, and sums, $f(a) + f(n - a)$, for $a$ up to some function of $n$. The rate-limiting factor is iterating through the sums. We discuss potential improvements to this algorithm via a method that provides a strong uniform bound on the number of summands that must be calculated for almost all $n$. We also develop code to run J. Arias de Reyna and J. van de Lune's analysis in higher bases and thus reduce their runtime of $\mathcal{O}(n^{1.230175})$ to $\mathcal{O}(n^{1.222911236})$. All of our code can be found online at: https://github.com/kcordwel/Integer-Complexity.

math.NT↗

arXiv2017

When almost all sets are difference dominated in $\mathbb{Z}/n\mathbb{Z}$

We investigate the behavior of the sum and difference sets of $A \subseteq \mathbb{Z}/n\mathbb{Z}$ chosen independently and randomly according to a binomial parameter $p(n) = o(1)$. We show that for rapidly decaying $p(n)$, $A$ is almost surely difference-dominated as $n \to \infty$, but for slowly decaying $p(n)$, $A$ is almost surely balanced as $n \to \infty$, with a continuous phase transition as $p(n)$ crosses a critical threshold. Specifically, we show that if $p(n) = o(n^{-1/2})$, then $|A-A|/|A+A|$ converges to $2$ almost surely as $n \to \infty$ and if $p(n) = c \cdot n^{-1/2}$, then $|A-A|/|A+A|$ converges to $1+\exp(-c^2/2)$ almost surely as $n \to \infty$. In these cases, we modify the arguments of Hegarty and Miller on subsets of $\mathbb{Z}$ to prove our results. When $\sqrt{\log n} \cdot n^{-1/2} = o(p(n))$, we prove that $|A-A| = |A+A| = n$ almost surely as $n \to \infty$ if some additional restrictions are placed on $n$. In this case, the behavior is drastically different from that of subsets of $\mathbb{Z}$ and new technical issues arise, so a novel approach is needed. When $n^{-1/2} = o(p(n))$ and $p(n) = o(\sqrt{ \log n} \cdot n^{-1/2})$, the behavior of $|A+A|$ and $|A-A|$ is markedly different and suggests an avenue for further study. These results establish a "correspondence principle" with the existing results of Hegarty, Miller, and Vissuet. As $p(n)$ decays more rapidly, the behavior of subsets of $\mathbb{Z}/n\mathbb{Z}$ approaches the behavior of subsets of $\mathbb{Z}$ shown by Hegarty and Miller. Moreover, as $p(n)$ decays more slowly, the behavior of subsets of $\mathbb{Z}/n\mathbb{Z}$ approaches the behavior shown by Miller and Vissuet in the case where $p(n) = 1/2$.

math.NT↗