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We consider a class of statistical estimation problems in which we are given a random data matrix ${\boldsymbol X}\in {\mathbb R}^{n\times d}$ (and possibly some labels ${\boldsymbol y}\in{\mathbb R}^n$) and would like to estimate a coefficient vector ${\boldsymbol \theta}\in{\mathbb R}^d$ (or possibly a constant number of such vectors). Special cases include low-rank matrix estimation and regularized estimation in generalized linear models (e.g., sparse regression). First order methods proceed by iteratively multiplying current estimates by ${\boldsymbol X}$ or its transpose. Examples include gradient descent or its accelerated variants. Celentano, Montanari, Wu proved that for any constant number of iterations (matrix vector multiplications), the optimal first order algorithm is a specific approximate message passing algorithm (known as `Bayes AMP'). The error of this estimator can be characterized in the high-dimensional asymptotics $n,d\to\infty$, $n/d\to\delta$, and provides a lower bound to the estimation error of any first order algorithm. Here we present a simpler proof of the same result, and generalize it to broader classes of data distributions and of first order algorithms, including algorithms with non-separable nonlinearities. Most importantly, the new proof technique does not require to construct an equivalent tree-structured estimation problem, and is therefore susceptible of a broader range of applications.

The Fr\'{e}chet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and study the Fr\'{e}chet distance for paths and walks on graphs. When provided with a distance oracle of $G$ with $O(1)$ query time, the classical quadratic-time dynamic program can compute the Fr\'{e}chet distance between two walks $P$ and $Q$ in a graph $G$ in $O(|P| \cdot |Q|)$ time. We show that there are situations where the graph structure helps with computing Fr\'{e}chet distance: when the graph $G$ is planar, we apply existing (approximate) distance oracles to compute a $(1+\varepsilon)$-approximation of the Fr\'{e}chet distance between any shortest path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{|Q|}{\varepsilon } )$ time. We generalise this result to near-shortest paths, i.e. $\kappa$-straight paths, as we show how to compute a $(1+\varepsilon)$-approximation between a $\kappa$-straight path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{\kappa|Q|}{\varepsilon } )$ time. Our algorithmic results hold for both the strong and the weak discrete Fr\'{e}chet distance over the shortest path metric in $G$. Finally, we show that additional assumptions on the input, such as our assumption on path straightness, are indeed necessary to obtain truly subquadratic running time. We provide a conditional lower bound showing that the Fr\'{e}chet distance, or even its $1.01$-approximation, between arbitrary \emph{paths} in a weighted planar graph cannot be computed in $O((|P|\cdot|Q|)^{1-\delta})$ time for any $\delta > 0$ unless the Orthogonal Vector Hypothesis fails. For walks, this lower bound holds even when $G$ is planar, unit-weight and has $O(1)$ vertices.

In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy version of the well-known graph isomorphism problem and appears in many applications, including social network deanonymization and cellular biology. Our focus here is on partial recovery, i.e., we look for a one-to-one mapping which is correct on a fraction of the nodes of the graph rather than on all of them, and we assume that the two input graphs to the problem are correlated Erd\H{o}s-R\'enyi graphs of parameters $(n,q,s)$. Our main contribution is then to give necessary and sufficient conditions on $(n,q,s)$ under which partial recovery is possible with high probability as the number of nodes $n$ goes to infinity. In particular, we show that it is possible to achieve partial recovery in the $nqs=\Theta(1)$ regime under certain additional assumptions.

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.

Sampling-based motion planning algorithms are widely used in robotics because they are very effective in high-dimensional spaces. However, the success rate and quality of the solutions are determined by an adequate selection of their parameters such as the distance between states, the local planner, and the sampling distribution. For robots with large configuration spaces or dynamic restrictions, selecting these parameters is a challenging task. This paper proposes a method for improving the performance to a set of the most popular sampling-based algorithms, the Rapidly-exploring Random Trees (RRTs) by adjusting the sampling method. The idea is to replace the uniform probability density function (U-PDF) with a custom distribution (C-PDF) learned from previously successful queries in similar tasks. With a few samples, our method builds a custom distribution that allows the RRT to grow to promising states that will lead to a solution. We tested our method in several autonomous driving tasks such as parking maneuvers, obstacle clearance and under narrow passages scenarios. The results show that the proposed method outperforms the original RRT and several improved versions in terms of success rate, tree density and computation time. In addition, the proposed method requires a relatively small set of examples, unlike current deep learning techniques that require a vast amount of examples.

Union-free codes and disjunctive codes are two combinatorial structures, which are used in nonadaptive group testing to find a set of $d$ defective elements among $n$ samples by carrying out the minimal number of tests $t$. It is known that union-free codes have a larger rate, whereas disjunctive codes provide a more efficient decoding algorithm. In this paper we introduce a new family of codes for nonadaptive group testing with fast decoding. The rate of these codes is larger than the rate of disjunctive codes, while the decoding algorithm has the same complexity. In addition, we derive a lower bound on the rate of new codes for the case of $d=2$ defectives, which is significantly better than the bound for disjunctive codes and almost as good as the bound for union-free codes.

We study regression discontinuity designs in which many covariates, possibly much more than the number of observations, are available. We consider a two-step algorithm which first selects the set of covariates to be used through a localized Lasso-type procedure, and then, in a second step, estimates the treatment effect by including the selected covariates into the usual local linear estimator. We provide an in-depth analysis of the algorithm's theoretical properties, showing that, under an approximate sparsity condition, the resulting estimator is asymptotically normal, with asymptotic bias and variance that are conceptually similar to those obtained in low-dimensional settings. Bandwidth selection and inference can be carried out using standard methods. We also provide simulations and an empirical application.

The current paper studies the problem of minimizing a loss $f(\boldsymbol{x})$ subject to constraints of the form $\boldsymbol{D}\boldsymbol{x} \in S$, where $S$ is a closed set, convex or not, and $\boldsymbol{D}$ is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method with the proximal distance principle. The latter is driven by minimization of penalized objectives $f(\boldsymbol{x})+\frac{\rho}{2}\text{dist}(\boldsymbol{D}\boldsymbol{x},S)^2$ involving large tuning constants $\rho$ and the squared Euclidean distance of $\boldsymbol{D}\boldsymbol{x}$ from $S$. The next iterate $\boldsymbol{x}_{n+1}$ of the corresponding proximal distance algorithm is constructed from the current iterate $\boldsymbol{x}_n$ by minimizing the majorizing surrogate function $f(\boldsymbol{x})+\frac{\rho}{2}\|\boldsymbol{D}\boldsymbol{x}-\mathcal{P}_{S}(\boldsymbol{D}\boldsymbol{x}_n)\|^2$. For fixed $\rho$ and a subanalytic loss $f(\boldsymbol{x})$ and a subanalytic constraint set $S$, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare against the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems.

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations.

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = \Tr(x)$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better minimum distance then than what the Goppa bound $d \geq 2deg(g(x))+1$ implies. Our improvement is based on finding another Goppa polynomial $h$ such that $C(L,g) = C(M, h)$ but $deg(h) > deg(g)$. This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case $m = 2$), as the Goppa bound for the quadratic case is sharp.