arXiv subjects

Rachana Madhukara

Publications and source records attributed to Rachana Madhukara.

7 recordsLinked to original sources

arXiv2026

The Radio-Frequency Transformer for Signal Separation

We study a problem of signal separation: estimating a signal of interest (SOI) contaminated by an unknown non-Gaussian background/interference. Given the training data consisting of examples of SOI and interference, we show how to build a fully data-driven signal separator. To that end we learn a good discrete tokenizer for SOI and then train an end-to-end transformer on a cross-entropy loss. Training with a cross-entropy shows substantial improvements over the conventional mean-squared error (MSE). Our tokenizer is a modification of Google's SoundStream, which incorporates additional transformer layers and switches from VQVAE to finite-scalar quantization (FSQ). Across real and synthetic mixtures from the MIT RF Challenge dataset, our method achieves competitive performance, including a 122x reduction in bit-error rate (BER) over prior state-of-the-art techniques for separating a QPSK signal from 5G interference. The learned representation adapts to the interference type without side information and shows zero-shot generalization to unseen mixtures at inference time, underscoring its potential beyond RF. Although we instantiate our approach on radio-frequency mixtures, we expect the same architecture to apply to gravitational-wave data (e.g., LIGO strain) and other scientific sensing problems that require data-driven modeling of background and noise.

cs.LG↗

arXiv2024

Adjacency and Broadcast Dimension of Grid and Directed Graphs

Let $G$ be a simple undirected graph. A function $f : V(G) \to \mathbb{Z}_{\geq 0}$ is a $\textit{resolving broadcast}$ of $G$ if for any distinct $x, y \in V(G)$, there exists a vertex $z \in V(G)$ with $f(z) > 0$ such that $\min \{ d(z, x), f(z)+1 \} \neq \min \{ d(z, y), f(z)+1 \}$. The $\textit{broadcast dimension}$ $\text{bdim}(G)$ of $G$ is the minimum of $\sum_{v \in V(G)} f(v)$ over all resolving broadcasts $f$ of $G$. Similarly, the $\textit{adjacency dimension}$ $\text{adim}(G)$ of $G$ is the minimum of $\sum_{v \in V(G)} f(v)$ over all resolving broadcasts $f$ of $G$ where $f$ takes values in $\{0,1\}$. These parameters are defined analogously for directed graphs by considering directed distances. We partially resolve a question of Zhang by obtaining precise bounds for the adjacency dimension of certain Cartesian products of path graphs, namely $\text{adim}(P_2 \square P_n)$ and $\text{adim}(P_3 \square P_n)$. Additionally, we study the behavior of adjacency and broadcast dimension on directed graphs. First, we explicitly calculate the adjacency dimension of a directed complete $k$-ary tree, where every edge is directed towards the leaves. Next, we prove that $\text{adim}(\vec{G}) = \text{bdim}(\vec{G})$ for some particular directed trees $\vec{G}$. Furthermore, we show that $\text{bdim}(G)$ can be as large as an exponential function of $\text{bdim}(\vec{G})$ or as small as a logarithmic function of $\text{bdim}(\vec{G})$.

math.CO↗

arXiv2023

Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins

The first author recently introduced toric promotion, an operator that acts on the labelings of a graph $G$ and serves as a cyclic analogue of Schützenberger's promotion operator. Toric promotion is defined as the composition of certain toggle operators, listed in a natural cyclic order. We consider more general permutoric promotion operators, which are defined as compositions of the same toggles, but in permuted orders. We settle a conjecture of the first author by determining the orders of all permutoric promotion operators when $G$ is a path graph. In fact, we completely characterize the orbit structures of these operators, showing that they satisfy the cyclic sieving phenomenon. The first half of our proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph.

math.CO↗

arXiv2023

Orthogonal Fold & Cut

We characterize the cut patterns that can be produced by "orthogonal fold & cut": folding an axis-aligned rectangular sheet of paper along horizontal and vertical creases, and then making a single straight cut (at any angle). Along the way, we solve a handful of related problems: orthogonal fold & punch, 1D fold & cut, signed 1D fold & cut, and 1D interval fold & cut.

cs.CG↗

arXiv2023

Improving the Runtime of Algorithmic Polarization of Hidden Markov Models

We improve the runtime of the linear compression scheme for hidden Markov sources presented in a 2018 paper of Guruswami, Nakkiran, and Sudan. Under the previous scheme, compressing a message of length $n$ takes $O(n \log n)$ runtime, and decompressing takes $O(n^{1 + δ})$ runtime for any fixed $δ> 0.$ We present how to improve the runtime of the decoding scheme to $O(n \log n)$ by caching intermediate results to avoid repeating computation.

cs.IT↗

arXiv2021

Edge-Unfolding Prismatoids: Tall or Rectangular Base

We show how to edge-unfold a new class of convex polyhedra, specifically a new class of prismatoids (the convex hull of two parallel convex polygons, called the top and base), by constructing a nonoverlapping "petal unfolding" in two new cases: (1) when the top and base are sufficiently far from each other; and (2) when the base is a rectangle and all other faces are nonobtuse triangles. The latter result extends a previous result by O'Rourke that the petal unfolding of a prismatoid avoids overlap when the base is a triangle (possibly obtuse) and all other faces are nonobtuse triangles. We also illustrate the difficulty of extending this result to a general quadrilateral base by giving a counterexample to our technique.

cs.CG↗

arXiv2021

Fourier Coefficients of Level 1 Hecke Eigenforms

Lehmer's 1947 conjecture on whether $τ(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $τ(n)$. For example, among the odd numbers $α$ such that $|α|<99$, we determine that $$τ(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $τ(n) \neq -|α|$ and that $τ(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $ΔE_4 = \sum_{n = 1}^{\infty} τ_{16}(n)q^n$, we show that \begin{align*}τ_{16}(n) \notin &\{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} \\ & \quad \quad \quad \quad \quad \cup \{-33,-55,-59,-67,-89,-91\}.\end{align*} Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.

math.NT↗