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A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential results. We prove a lower bound of $\Omega(2^{n/2})$ for the number of queries needed to test whether a Boolean function $f:\mathbb{F}_2^n \to \{0,1\}$ is the indicator function of a sumset. Our lower bound for testing sumsets follows from sharp bounds on the related problem of shift testing, which may be of independent interest. We also give a near-optimal $2^{n/2} \cdot \mathrm{poly}(n)$-query algorithm for a smoothed analysis formulation of the sumset refutation problem.

Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $\vec{m}_\text{src}$ cannot reach a marking $\vec{m}_\text{tgt}$, then there is a formula $\varphi$ of Presburger arithmetic such that: $\varphi(\vec{m}_\text{src})$ holds; $\varphi$ is forward invariant, i.e., $\varphi(\vec{m})$ and $\vec{m} \rightarrow \vec{m}'$ imply $\varphi(\vec{m}'$); and $\neg \varphi(\vec{m}_\text{tgt})$ holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their worst-case size, which is at least Ackermannian, and the complexity of checking that a formula is a separator, which is at least exponential (in the formula size). We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachability, which is P-complete). We further consider the more general problem of (existential) set-to-set reachability, where two sets of markings are given as convex polytopes. We show that, while our approach does not extend directly, we can efficiently certify unreachability via an altered Petri net.

Recently, an interesting phenomenon called grokking has gained much attention, where generalization occurs long after the models have initially overfitted the training data. We try to understand this seemingly strange phenomenon through the robustness of the neural network. From a robustness perspective, we show that the popular $l_2$ weight norm (metric) of the neural network is actually a sufficient condition for grokking. Based on the previous observations, we propose perturbation-based methods to speed up the generalization process. In addition, we examine the standard training process on the modulo addition dataset and find that it hardly learns other basic group operations before grokking, for example, the commutative law. Interestingly, the speed-up of generalization when using our proposed method can be explained by learning the commutative law, a necessary condition when the model groks on the test dataset. We also empirically find that $l_2$ norm correlates with grokking on the test data not in a timely way, we propose new metrics based on robustness and information theory and find that our new metrics correlate well with the grokking phenomenon and may be used to predict grokking.

We introduce a test for the conditional independence of random variables $X$ and $Y$ given a random variable $Z$, specifically by sampling from the joint distribution $(X,Y,Z)$, binning the support of the distribution of $Z$, and conducting multiple $p$-Wasserstein two-sample tests. Under a $p$-Wasserstein Lipschitz assumption on the conditional distributions $\mathcal{L}_{X|Z}$, $\mathcal{L}_{Y|Z}$, and $\mathcal{L}_{(X,Y)|Z}$, we show that it is possible to control the Type I and Type II error of this test, and give examples of explicit finite-sample error bounds in the case where the distribution of $Z$ has compact support.

Let $\mathcal{H}=(X,\mathcal{E})$ be a hypergraph. A support is a graph $Q$ on $X$ such that for each $E\in\mathcal{E}$, the subgraph of $Q$ induced on the elements in $E$ is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph $G=(V,E)$, with $c:V\to\{\mathbf{r},\mathbf{b}\}$, and a collection of connected subgraphs $\mathcal{H}$ of $G$, a primal support is a graph $Q$ on $\mathbf{b}(V)$ such that for each $H\in \mathcal{H}$, the induced subgraph $Q[\mathbf{b}(H)]$ on vertices $\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b})$ is connected. A \emph{dual support} is a graph $Q^*$ on $\mathcal{H}$ s.t. for each $v\in X$, the induced subgraph $Q^*[\mathcal{H}_v]$ is connected, where $\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}$. We present sufficient conditions on the host graph and hyperedges so that the resulting support comes from a restricted family. We primarily study two classes of graphs: $(1)$ If the host graph has genus $g$ and the hypergraphs satisfy a topological condition of being \emph{cross-free}, then there is a primal and a dual support of genus at most $g$. $(2)$ If the host graph has treewidth $t$ and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth $O(2^t)$. We show that this exponential blow-up is sometimes necessary. As an intermediate case, we also study the case when the host graph is outerplanar. Finally, we show applications of our results to packing and covering, and coloring problems on geometric hypergraphs.

Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(\sigma) = (f(\sigma_1), \ldots, f(\sigma_k))$ for each $k$-face $\sigma$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $\delta>0$, there exists a family of high-dimensional expanders with degree $O_{\delta}(1)$ and a $2$-query direct product tester with soundness $\delta$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness.

A $d$-dimensional simplicial complex $X$ is said to support a direct product tester if any locally consistent function defined on its $k$-faces (where $k\ll d$) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution $\mu$ over pairs of $k$-faces $(A,A')$, and given query access to $F\colon X(k)\to\{0,1\}^k$ it samples $(A,A')\sim \mu$ and checks that $F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$. The tester should have (1) the ``completeness property'', meaning that any assignment $F$ which is a direct product assignment passes the test with probability $1$, and (2) the ``soundness property'', meaning that if $F$ passes the test with probability $s$, then $F$ must be correlated with a direct product function. Dinur and Kaufman showed that a sufficiently good spectral expanding complex $X$ admits a direct product tester in the ``high soundness'' regime where $s$ is close to $1$. They asked whether there are high dimensional expanders that support direct product tests in the ``low soundness'', when $s$ is close to $0$. We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as \emph{Unique-Games coboundary expanders}. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution.

We give an example of a class of distributions that is learnable in total variation distance with a finite number of samples, but not learnable under $(\varepsilon, \delta)$-differential privacy. This refutes a conjecture of Ashtiani.

In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.

If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups.