A Young diagram $Y$ is called wide if every sub-diagram $Z$ formed by a subset of the rows of $Y$ dominates $Z'$, the conjugate of $Z$. A Young diagram $Y$ is called Latin if its squares can be assigned numbers so that for each $i$, the $i$th row is filled injectively with the numbers $1, \ldots ,a_i$, where $a_i$ is the length of $i$th row of $Y$, and every column is also filled injectively. A conjecture of Chow and Taylor, publicized by Chow, Fan, Goemans, and Vondrak is that a wide Young diagram is Latin. We prove a dual version of the conjecture.
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The Strong Exponential Hierarchy $SEH$ was shown to collapse to $P^{NExp}$ by Hemachandra by proving $P^{NExp} = NP^{NExp}$ via a census argument. Nonetheless, Hemachandra also asked for certificate-based and alternating Turing machine characterizations of the $SEH$ levels, in the hope that these might have revealed deeper structural reasons behind the collapse. These open questions have thus far remained unanswered. To close them, by building upon the notion of Hausdorff reductions, we investigate a natural normal form for the intermediate levels of the (generalized) exponential hierarchies, i.e., the single-, the double-Exponential Hierarchy, and so on. Although the two characterizations asked for derive from our Hausdorff characterization, it is nevertheless from the latter that a surprising structural reason behind the collapse of $SEH$ is uncovered as a consequence of a very general result: the intermediate levels of the exponential hierarchies are precisely characterized by specific "Hausdorff classes", which define these levels without resorting to oracle machines. By this, contrarily to oracle classes, which may have different shapes for a same class (e.g., $P^{NP}_{||} = P^{NP[Log]} = LogSpace^{NP}$), hierarchy intermediate levels are univocally identified by Hausdorff classes (under the hypothesis of no hierarchy collapse). In fact, we show that the rather simple reason behind many equivalences of oracle classes is that they just refer to different ways of deciding the languages of a same Hausdorff class, and this happens also for $P^{NExp}$ and $NP^{NExp}$. In addition, via Hausdorff classes, we define complete problems for various intermediate levels of the exponential hierarchies. Through these, we obtain matching lower-bounds for problems known to be in $P^{NExp[Log]}$, but whose hardness was left open due to the lack of known $P^{NExp[Log]}$-complete problems.
The solution to empirical risk minimization with $f$-divergence regularization (ERM-$f$DR) is presented under mild conditions on $f$. Under such conditions, the optimal measure is shown to be unique. Examples of the solution for particular choices of the function $f$ are presented. Previously known solutions to common regularization choices are obtained by leveraging the flexibility of the family of $f$-divergences. These include the unique solutions to empirical risk minimization with relative entropy regularization (Type-I and Type-II). The analysis of the solution unveils the following properties of $f$-divergences when used in the ERM-$f$DR problem: $i\bigl)$ $f$-divergence regularization forces the support of the solution to coincide with the support of the reference measure, which introduces a strong inductive bias that dominates the evidence provided by the training data; and $ii\bigl)$ any $f$-divergence regularization is equivalent to a different $f$-divergence regularization with an appropriate transformation of the empirical risk function.
Let $S_d(n)$ denote the minimum number of wires of a depth-$d$ (unbounded fan-in) circuit encoding an error-correcting code $C:\{0, 1\}^n \to \{0, 1\}^{32n}$ with distance at least $4n$. G\'{a}l, Hansen, Kouck\'{y}, Pudl\'{a}k, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] proved that $S_d(n) = \Theta_d(\lambda_d(n)\cdot n)$ for any fixed $d \ge 3$. By improving their construction and analysis, we prove $S_d(n)= O(\lambda_d(n)\cdot n)$. Letting $d = \alpha(n)$, a version of the inverse Ackermann function, we obtain circuits of linear size. This depth $\alpha(n)$ is the minimum possible to within an additive constant 2; we credit the nearly-matching depth lower bound to G\'{a}l et al., since it directly follows their method (although not explicitly claimed or fully verified in that work), and is obtained by making some constants explicit in a graph-theoretic lemma of Pudl\'{a}k [Combinatorica, 14(2), 1994], extending it to super-constant depths. We also study a subclass of MDS codes $C: \mathbb{F}^n \to \mathbb{F}^m$ characterized by the Hamming-distance relation $\mathrm{dist}(C(x), C(y)) \ge m - \mathrm{dist}(x, y) + 1$ for any distinct $x, y \in \mathbb{F}^n$. (For linear codes this is equivalent to the generator matrix being totally invertible.) We call these superconcentrator-induced codes, and we show their tight connection with superconcentrators. Specifically, we observe that any linear or nonlinear circuit encoding a superconcentrator-induced code must be a superconcentrator graph, and any superconcentrator graph can be converted to a linear circuit, over a sufficiently large field (exponential in the size of the graph), encoding a superconcentrator-induced code.
Assuming we have iid observations from two unknown probability density functions (pdfs), $p$ and $q$, the likelihood-ratio estimation (LRE) is an elegant approach to compare the two pdfs only by relying on the available data. In this paper, we introduce the first -to the best of our knowledge-graph-based extension of this problem, which reads as follows: Suppose each node $v$ of a fixed graph has access to observations coming from two unknown node-specific pdfs, $p_v$ and $q_v$, and the goal is to estimate for each node the likelihood-ratio between both pdfs by also taking into account the information provided by the graph structure. The node-level estimation tasks are supposed to exhibit similarities conveyed by the graph, which suggests that the nodes could collaborate to solve them more efficiently. We develop this idea in a concrete non-parametric method that we call Graph-based Relative Unconstrained Least-squares Importance Fitting (GRULSIF). We derive convergence rates for our collaborative approach that highlights the role played by variables such as the number of available observations per node, the size of the graph, and how accurately the graph structure encodes the similarity between tasks. These theoretical results explicit the situations where collaborative estimation effectively leads to an improvement in performance compared to solving each problem independently. Finally, in a series of experiments, we illustrate how GRULSIF infers the likelihood-ratios at the nodes of the graph more accurately compared to state-of-the art LRE methods, which would operate independently at each node, and we also verify that the behavior of GRULSIF is aligned with our previous theoretical analysis.
The problem of estimating a parameter in the drift coefficient is addressed for $N$ discretely observed independent and identically distributed stochastic differential equations (SDEs). This is done considering additional constraints, wherein only public data can be published and used for inference. The concept of local differential privacy (LDP) is formally introduced for a system of stochastic differential equations. The objective is to estimate the drift parameter by proposing a contrast function based on a pseudo-likelihood approach. A suitably scaled Laplace noise is incorporated to meet the privacy requirements. Our key findings encompass the derivation of explicit conditions tied to the privacy level. Under these conditions, we establish the consistency and asymptotic normality of the associated estimator. Notably, the convergence rate is intricately linked to the privacy level, and is some situations may be completely different from the case where privacy constraints are ignored. Our results hold true as the discretization step approaches zero and the number of processes $N$ tends to infinity.
This paper addresses the challenge of the private information retrieval (PIR) problem wherein there are $N$ replicated non-communicating databases containing the same $M$ messages and a user who wants to retrieve one of the messages without revealing the wanted message's index to the databases. In addition, we assume a block-fading additive white Gaussian noise multiple access channel (AWGN MAC) linking the user and the databases. Shmuel's contribution \cite{shmuel2021private}, presenting a joint channel-PIR scheme utilizing the C\&F protocol, has shown the potential of a joint channel-PIR scheme over a separated scheme. In this paper, we propose an improved joint channel-PIR approach tailored for the PIR problem with $N$ databases over a block-fading AWGN. Unlike the C\&F protocol, our scheme offers reduced computational complexity while improving the scaling laws governing the achievable rate. Our achievable rate scales with the number of databases $N$ and the power $P$ similarly to the channel capacity without the privacy constraint and outperforms the C\&F-based approach. Furthermore, our analysis demonstrates that our improved rate exhibits only a finite gap from the channel capacity of one bit as $N$ increases.
The forcing number of a graph with a perfect matching $M$ is the minimum number of edges in $M$ whose endpoints need to be deleted, such that the remaining graph only has a single perfect matching. This number is of great interest in theoretical chemistry, since it conveys information about the structural properties of several interesting molecules. On the other hand, in bipartite graphs the forcing number corresponds to the famous feedback vertex set problem in digraphs. Determining the complexity of finding the smallest forcing number of a given planar graph is still a widely open and important question in this area, originally proposed by Afshani, Hatami, and Mahmoodian in 2004. We take a first step towards the resolution of this question by providing an algorithm that determines the set of all possible forcing numbers of an outerplanar graph in polynomial time. This is the first polynomial-time algorithm concerning this problem for a class of graphs of comparable or greater generality.
In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and $F=\mathbb{F}_q$, a self-dual MRD code exists if and only if $q\equiv n\equiv 3 \ [4].$
Simulating mixtures of distributions with signed weights proves a challenge as standard simulation algorithms are inefficient in handling the negative weights. In particular, the natural representation of mixture variates as associated with latent component indicators is no longer available. We propose here an exact accept-reject algorithm in the general case of finite signed mixtures that relies on optimaly pairing positive and negative components and designing a stratified sampling scheme on pairs. We analyze the performances of our approach, relative to the inverse cdf approach, since the cdf of the distribution remains available for standard signed mixtures.
Consider the task of estimating a random vector $X$ from noisy observations $Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore, for $p > 2$, it is demonstrated that there are infinitely many priors that can induce such an estimator.
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