We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$
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In this paper we study the random geometric graph $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ with $L_q$ distance where each vertex is sampled uniformly from the $d$-dimensional torus and where the connection radius is chosen so that the marginal edge probability is $p$. In addition to results addressing other questions, we make progress on determining when it is possible to distinguish $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ from the Endos-R\'enyi graph $\mathsf{G}(n,p)$. Our strongest result is in the extreme setting $q = \infty$, in which case $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^\infty_p,p)$ is the $\mathsf{AND}$ of $d$ 1-dimensional random geometric graphs. We derive a formula similar to the cluster-expansion from statistical physics, capturing the compatibility of subgraphs from each of the $d$ 1-dimensional copies, and use it to bound the signed expectations of small subgraphs. We show that counting signed 4-cycles is optimal among all low-degree tests, succeeding with high probability if and only if $d = \tilde{o}(np).$ In contrast, the signed triangle test is suboptimal and only succeeds when $d = \tilde{o}((np)^{3/4}).$ Our result stands in sharp contrast to the existing literature on random geometric graphs (mostly focused on $L_2$ geometry) where the signed triangle statistic is optimal.
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a smooth number, then the divide-and-conquer approach leads to the fastest known FFT algorithms. Depending on the type of group that the set of evaluation points forms, these algorithms are classified as multiplicative (Math of Comp. 1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a unified framework for FFT algorithms that include both multiplicative and additive FFT algorithms as special cases, and beyond: our framework also works when $q+1$ is smooth, while all known results require $q$ or $q-1$ to be smooth. For the new case where $q+1$ is smooth (this new case was not considered before in literature as far as we know), we show that if $n$ is a divisor of $q+1$ that is $B$-smooth for a real $B>0$, then our FFT needs $O(Bn\log n)$ arithmetic operations in $\mathbb{F}_q$. Our unified framework is a natural consequence of introducing the algebraic function fields into the study of FFT.
We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying software to further inspect the polytope of an input graph is available.
Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$ The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$. Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before. Moreover, the residue codes of the constructed extremal $\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes form self-orthogonal $1$-designs.
A periodic temporal graph $\mathcal{G}=(G_0, G_1, \dots, G_{p-1})^*$ is an infinite periodic sequence of graphs $G_i=(V,E_i)$ where $G=(V,\cup_i E_i)$ is called the footprint. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the copnumber is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the copnumber $c(\mathcal{G})$ of a periodic graph $\mathcal{G}$ and the copnumber $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c(\mathcal{G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the copnumbers of a periodic graph $\mathcal{G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the copnumber of a periodic graph from properties of its footprint such as its treewidth.
We give a strongly explicit construction of $\varepsilon$-approximate $k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group $\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality $\mathrm{poly}(N^k/\varepsilon)$ (equivalently, they have seed length $O(nk + \log(1/\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.
A code $C \subseteq \{0, 1, 2\}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T(n)$ we denote the maximum cardinality of trifferent codes with length. $T(5)=10$ and $T(6)=13$ were recently determined. Here we determine $T(7)=16$, $T(8)=20$, and $T(9)=27$. For the latter case $n=9$ there also exist linear codes attaining the maximum possible cardinality $27$.
Austrin showed that the approximation ratio $\beta\approx 0.94016567$ obtained by the MAX 2-SAT approximation algorithm of Lewin, Livnat and Zwick (LLZ) is optimal modulo the Unique Games Conjecture (UGC) and modulo a Simplicity Conjecture that states that the worst performance of the algorithm is obtained on so called simple configurations. We prove Austrin's conjecture, thereby showing the optimality of the LLZ approximation algorithm, relying only on the Unique Games Conjecture. Our proof uses a combination of analytic and computational tools. We also present new approximation algorithms for two restrictions of the MAX 2-SAT problem. For MAX HORN-$\{1,2\}$-SAT, i.e., MAX CSP$(\{x\lor y,\bar{x}\lor y,x,\bar{x}\})$, in which clauses are not allowed to contain two negated literals, we obtain an approximation ratio of $0.94615981$. For MAX CSP$(\{x\lor y,x,\bar{x}\})$, i.e., when 2-clauses are not allowed to contain negated literals, we obtain an approximation ratio of $0.95397990$. By adapting Austrin's and our arguments for the MAX 2-SAT problem we show that these two approximation ratios are also tight, modulo only the UGC conjecture. This completes a full characterization of the approximability of the MAX 2-SAT problem and its restrictions.
We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.
We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with $\exp(O(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was $\Omega(\sqrt{\exp(d)/\varepsilon})$ in the small $\varepsilon$ parameter regime, when $\varepsilon = O(d^{-3/2})$, due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for $\varepsilon \gg d^{-3/2}$. We resolve this question, obtaining a tight lower bound of $\exp(\Omega(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ for all $\varepsilon$ at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of $k$-monotonicity testing and learning for functions $f \colon \{0,1\}^d \to [r]$ is $\exp(\Theta(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$. For testing with one-sided error we show that the sample complexity is $\exp(\Theta(d))$. Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of $d,k,r,1/\varepsilon$ in the exponent) of $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$ on the sample complexity of testing and learning measurable $k$-monotone functions $f \colon \mathbb{R}^d \to [r]$ under product distributions. Our upper bound improves upon the previous bound of $\exp(\widetilde{O}(\min\{\frac{k}{\varepsilon^2}\sqrt{d},d\}))$ by Harms-Yoshida (ICALP 2022) for Boolean functions ($r=2$).
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