arXiv subjects

Neil Krishnan

Publications and source records attributed to Neil Krishnan.

3 recordsLinked to original sources

arXiv2026

Depth Lower Bounds for ReLU Networks with Binary Inputs

We study the role of depth in ReLU networks with discrete (Boolean) inputs and real-valued outputs, complementing two established lines of work. For Boolean inputs, striking depth separation results were proven for $\mathsf{AC}^0$ but with threshold ($\mathsf{TC}^0$) or ReLU gates depth separation is only established for depth two vs. three. On the other hand, for {\em real-valued} functions and ReLU networks, Telgarsky's (2016) constructed a simple one variable class of functions which establishes separation at higher depths. In this paper we are interested to establish an all-depths depth separation for ReLU networks on $\{0,1\}^n$. We do so by exhibiting an explicit family of functions computable exactly by a ReLU network of depth $n+1$ and constant width, such that any ReLU network of depth $d$ and width $w$ computing the function exactly must satisfy $w^d = \Omega(2^n)$; in particular, no network of depth $d = o(n/\log n)$ can compute it with width polynomial in $n$. We note that our lower bound relies on \emph{exact, infinite-accuracy} computation as an exponential precision truncation of the output is computable by a polynomial-size $\mathsf{TC}^0$ circuit.

cs.CC↗

arXiv2024

Vertex Connectivity of Friends-and-strangers Graphs

Friends-and-strangers graphs, coined by Defant and Kravitz, are denoted by $\mathsf{FS}(X,Y)$ where $X$ and $Y$ are both graphs on $n$ vertices. The graph $X$ represents positions and edges mark adjacent positions while the graph $Y$ represents people and edges mark friendships. The vertex set of $\mathsf{FS}(X,Y)$ consists of all one-to-one placements of people on positions, and there is an edge between any two placements if it is possible to swap two people who are friends and on adjacent positions to get from one placement to the other. Previous papers have studied when $\mathsf{FS}(X,Y)$ is connected. In this paper, we consider when $\mathsf{FS}(X,Y)$ is $k$-connected where a graph is $k$-connected if it remains connected after removing any $k-1$ or less vertices. We first consider $\mathsf{FS}(X,Y)$ when $Y$ is a complete graph or star graph. We find tight bounds on their connectivity, proving their connectivity equals their minimum degree. We further consider the size of the connected components of $\mathsf{FS}(X,\mathsf{Star}_n)$ where $X$ is connected. We show that asymptotically similar conditions as the conditions mentioned by Bangachev are sufficient for $\mathsf{FS}(X,Y)$ to be $k$-connected. Finally, we consider when $X$ and $Y$ are independent Erd\H{o}s--R\'enyi random graphs on $n$ vertices and edge probability $p_1$ and $p_2,$ respectively. We show that for $p_0 = n^{-1/2+o(1)},$ if $p_1p_2\geq p_0^2$ and $p_1,$ $p_2 \geq w(n) p_0$ where $w(n) \rightarrow 0$ as $n \rightarrow \infty,$ then $\mathsf{FS}(X,Y)$ is $k$-connected with high probability. This is asymptotically tight as we show that below an asymptotically similar threshold $p_0'=n^{-1/2+o(1)}$, the graph $\mathsf{FS}(X,Y)$ is disconnected with high probability if $p_1p_2 \leq (p_0')^2$.

math.CO↗

arXiv2024

Multiplayer Games of War

A recent paper by Bhatia, Chin, Mani, and Mossel (2026) defined stochastic processes modeling the game of War for {\em two players} with $n$ cards. That paper showed that these models, assuming uniform random decks, are equivalent to the Gambler's Ruin problem and therefore have an expected termination time of $\Theta(n^2)$. In this paper, we generalize these models to {\em any number of players} $m$. We prove that the game with $m$ players is equivalent to a simple sticky random walk on an $(m-1)$-simplex; therefore, the termination time is the same as the absorption time of the simple sticky random walk. Unlike Gambler's Ruin, this absorption time has not been analyzed before. We show that the absorption time of the walk and the termination time of the game are both $\Theta(n^2)$ for any number of players.

math.PR↗