Given an arbitrary set of high dimensional points in $\ell_1$, there are known negative results that preclude the possibility of always mapping them to a low dimensional $\ell_1$ space while preserving distances with small multiplicative distortion. This is in stark contrast with dimension reduction in Euclidean space ($\ell_2$) where such mappings are always possible. While the first non-trivial lower bounds for $\ell_1$ dimension reduction were established almost 20 years ago, there has been limited progress in understanding what sets of points in $\ell_1$ are conducive to a low-dimensional mapping. In this work, we study a new characterization of $\ell_1$ metrics that are conducive to dimension reduction in $\ell_1$. Our characterization focuses on metrics that are defined by the disagreement of binary variables over a probability distribution -- any $\ell_1$ metric can be represented in this form. We show that, for configurations of $n$ points in $\ell_1$ obtained from tree Ising models, we can reduce dimension to $\mathrm{polylog}(n)$ with constant distortion. In doing so, we develop technical tools for embedding truncated metrics which have been studied because of their applications in computer vision, and are objects of independent interest in metric geometry. Among other tools, we show how any $\ell_1$ metric can be truncated with $O(1)$ distortion and $O(\log(n))$ blowup in dimension.
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Differentially private computation often begins with a bound on some $d$-dimensional statistic's $\ell_p$ sensitivity. For pure differential privacy, the $K$-norm mechanism can improve on this approach using a norm tailored to the statistic's sensitivity space. Writing down a closed-form description of this optimal norm is often straightforward. However, running the $K$-norm mechanism reduces to uniformly sampling the norm's unit ball; this ball is a $d$-dimensional convex body, so general sampling algorithms can be slow. Turning to concentrated differential privacy, elliptic Gaussian noise offers similar improvement over spherical Gaussian noise. Once the shape of this ellipse is determined, sampling is easy; however, identifying the best such shape may be hard. This paper solves both problems for the simple statistics of sum, count, and vote. For each statistic, we provide a sampler for the optimal $K$-norm mechanism that runs in time $\tilde O(d^2)$ and derive a closed-form expression for the optimal shape of elliptic Gaussian noise. The resulting algorithms all yield meaningful accuracy improvements while remaining fast and simple enough to be practical. More broadly, we suggest that problem-specific sensitivity space analysis may be an overlooked tool for private additive noise.
This article presents MAPS$^2$ : a distributed algorithm that allows multi-robot systems to deliver coupled tasks expressed as Signal Temporal Logic (STL) constraints. Classical control theoretical tools addressing STL constraints either adopt a limited fragment of the STL formula or require approximations of min/max operators, whereas works maximising robustness through optimisation-based methods often suffer from local minima, relaxing any completeness arguments due to the NP-hard nature of the problem. Endowed with probabilistic guarantees, MAPS$^2$ provides an anytime algorithm that iteratively improves the robots' trajectories. The algorithm selectively imposes spatial constraints by taking advantage of the temporal properties of the STL. The algorithm is distributed, in the sense that each robot calculates its trajectory by communicating only with its immediate neighbours as defined via a communication graph. We illustrate the efficiency of MAPS$^2$ by conducting extensive simulation and experimental studies, verifying the generation of STL satisfying trajectories.
M${}^{\natural}$-concave functions, a.k.a. gross substitute valuation functions, play a fundamental role in many fields, including discrete mathematics and economics. In practice, perfect knowledge of M${}^{\natural}$-concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M${}^{\natural}$-concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). For the stochastic bandit setting, we present $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms under $T$ times access to unbiased noisy value oracles of M${}^{\natural}$-concave functions. A key to proving these results is the robustness of the greedy algorithm to local errors in M${}^{\natural}$-concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve $O(T^{1-c})$ regret for any constant $c > 0$ unless $\mathsf{P} = \mathsf{NP}$. Our proof is based on a reduction from the matroid intersection problem for three matroids, which would be a novel idea in the context of online learning.
Fairness in decision-making processes is often quantified using probabilistic metrics. However, these metrics may not fully capture the real-world consequences of unfairness. In this article, we adopt a utility-based approach to more accurately measure the real-world impacts of decision-making process. In particular, we show that if the concept of $\varepsilon$-fairness is employed, it can possibly lead to outcomes that are maximally unfair in the real-world context. Additionally, we address the common issue of unavailable data on false negatives by proposing a reduced setting that still captures essential fairness considerations. We illustrate our findings with two real-world examples: college admissions and credit risk assessment. Our analysis reveals that while traditional probability-based evaluations might suggest fairness, a utility-based approach uncovers the necessary actions to truly achieve equality. For instance, in the college admission case, we find that enhancing completion rates is crucial for ensuring fairness. Summarizing, this paper highlights the importance of considering the real-world context when evaluating fairness.
We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$. The motivation for this problem lies in a recent construction for subspace codes based on cellular automata. More precisely, the minimum distance of such subspace codes relates to the maximum degree $d$ of the pairwise GCD in this family of polynomials. Hence, characterizing the maximal families of such polynomials is equivalent to determining the maximum cardinality of the corresponding subspace codes for a given minimum distance. We first show a lower bound on the cardinality of such families, and then focus on the specific case where $d=1$. There, we characterize the maximal families of polynomials over the binary field $\mathbb{F}_2$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.
Tensors serve as a crucial tool in the representation and analysis of complex, multi-dimensional data. As data volumes continue to expand, there is an increasing demand for developing optimization algorithms that can directly operate on tensors to deliver fast and effective computations. Many problems in real-world applications can be formulated as the task of recovering high-order tensors characterized by sparse and/or low-rank structures. In this work, we propose novel Kaczmarz algorithms with a power of the $\ell_1$-norm regularization for reconstructing high-order tensors by exploiting sparsity and/or low-rankness of tensor data. In addition, we develop both a block and an accelerated variant, along with a thorough convergence analysis of these algorithms. A variety of numerical experiments on both synthetic and real-world datasets demonstrate the effectiveness and significant potential of the proposed methods in image and video processing tasks, such as image sequence destriping and video deconvolution.
We show that computing the total variation distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.
We present fully abstract encodings of the call-by-name and call-by-value $\lambda$-calculus into HOcore, a minimal higher-order process calculus with no name restriction. We consider several equivalences on the $\lambda$-calculus side -- normal-form bisimilarity, applicative bisimilarity, and contextual equivalence -- that we internalize into abstract machines in order to prove full abstraction of the encodings. We also demonstrate that this technique scales to the $\lambda\mu$-calculus, i.e., a standard extension of the $\lambda$-calculus with control operators.
The problem of computing $\alpha$-capacity for $\alpha>1$ is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu--Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the $\alpha$-capacity for $\alpha>1$ based on a variational characterization of the Augustin--Csisz{\'a}r mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.
For a fixed positive integer $d \geq 2$, a distance-$d$ independent set (D$d$IS) of a graph is a vertex subset whose distance between any two members is at least $d$. Imagine that there is a token placed on each member of a D$d$IS. Two D$d$ISs are adjacent under Token Sliding ($\mathsf{TS}$) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping ($\mathsf{TJ}$), the target vertex needs not to be adjacent to the original one. The Distance-$d$ Independent Set Reconfiguration (D$d$ISR) problem under $\mathsf{TS}/\mathsf{TJ}$ asks if there is a corresponding sequence of adjacent D$d$ISs that transforms one given D$d$IS into another. The problem for $d = 2$, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of D$d$ISR on different graphs under $\mathsf{TS}$ and $\mathsf{TJ}$ for any fixed $d \geq 3$. On chordal graphs, we show that D$d$ISR under $\mathsf{TJ}$ is in $\mathtt{P}$ when $d$ is even and $\mathtt{PSPACE}$-complete when $d$ is odd. On split graphs, there is an interesting complexity dichotomy: D$d$ISR is $\mathtt{PSPACE}$-complete for $d = 2$ but in $\mathtt{P}$ for $d=3$ under $\mathsf{TS}$, while under $\mathsf{TJ}$ it is in $\mathtt{P}$ for $d = 2$ but $\mathtt{PSPACE}$-complete for $d = 3$. Additionally, certain well-known hardness results for $d = 2$ on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for $d \geq 3$.
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