The computation of the distance of two time series is time-consuming for any elastic distance function that accounts for misalignments. Among those functions, DTW is the most prominent. However, a recent extensive evaluation has shown that the move-split merge (MSM) metric is superior to DTW regarding the analytical accuracy of the 1-NN classifier. Unfortunately, the running time of the standard dynamic programming algorithm for MSM distance computation is $\Omega(n^2)$, where $n$ is the length of the longest time series. In this paper, we provide approaches to reducing the cost of MSM distance computations by using lower and upper bounds for early pruning paths in the underlying dynamic programming table. For the case of one time series being a constant, we present a linear-time algorithm. In addition, we propose new linear-time heuristics and adapt heuristics known from DTW to computing the MSM distance. One heuristic employs the metric property of MSM and the previously introduced linear-time algorithm. Our experimental studies demonstrate substantial speed-ups in our approaches compared to previous MSM algorithms. In particular, the running time for MSM is faster than a state-of-the-art DTW distance computation for a majority of the popular UCR data sets.
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Recently it has been shown that using diffusion models for inverse problems can lead to remarkable results. However, these approaches require a closed-form expression of the degradation model and can not support complex degradations. To overcome this limitation, we propose a method (INDigo) that combines invertible neural networks (INN) and diffusion models for general inverse problems. Specifically, we train the forward process of INN to simulate an arbitrary degradation process and use the inverse as a reconstruction process. During the diffusion sampling process, we impose an additional data-consistency step that minimizes the distance between the intermediate result and the INN-optimized result at every iteration, where the INN-optimized image is composed of the coarse information given by the observed degraded image and the details generated by the diffusion process. With the help of INN, our algorithm effectively estimates the details lost in the degradation process and is no longer limited by the requirement of knowing the closed-form expression of the degradation model. Experiments demonstrate that our algorithm obtains competitive results compared with recently leading methods both quantitatively and visually. Moreover, our algorithm performs well on more complex degradation models and real-world low-quality images.
The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.
We define the complexity of a continuous-time linear system to be the minimum number of bits required to describe its forward increments to a desired level of fidelity, and compute this quantity using the rate distortion function of a Gaussian source of uncertainty in those increments. The complexity of a linear system has relevance in control-communications contexts requiring local and dynamic decision-making based on sampled data representations. We relate this notion of complexity to the design of attention-varying controllers, and demonstrate a novel methodology for constructing source codes via the endpoint maps of so-called emulating systems, with potential for non-parametric, data-based simulation and analysis of unknown dynamical systems.
Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical problems here are to find a maxcore (subgraph of maximum min degree) and to find a densest subgraph (subgraph of maximum average degree). Both of these problems can be solved in polynomial time. Veldt, Benson, and Kleinberg [VBK21] introduced the generalized $p$-mean densest subgraph problem which captures the maxcore problem when $p=-\infty$ and the densest subgraph problem when $p=1$. They observed that the objective leads to a supermodular function when $p \ge 1$ and hence can be solved in polynomial time; for this case, they also developed a simple greedy peeling algorithm with a bounded approximation ratio. In this paper, we make several contributions. First, we prove that for any $p \in (-\frac{1}{8}, 0) \cup (0, \frac{1}{4})$ the problem is NP-Hard and for any $p \in (-3,0) \cup (0,1)$ the weighted version of the problem is NP-Hard, partly resolving a question left open in [VBK21]. Second, we describe two simple $1/2$-approximation algorithms for all $p < 1$, and show that our analysis of these algorithms is tight. For $p > 1$ we develop a fast near-linear time implementation of the greedy peeling algorithm from [VBK21]. This allows us to plug it into the iterative peeling algorithm that was shown to converge to an optimum solution [CQT22]. We demonstrate the efficacy of our algorithms by running extensive experiments on large graphs. Together, our results provide a comprehensive understanding of the complexity of the $p$-mean densest subgraph problem and lead to fast and provably good algorithms for the full range of $p$.
It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for multiplying two $2\times 2$ matrices in 7 instead of 8 multiplications. This gives rise to the constraint satisfaction problem of fast matrix multiplication, where a set of $R < NMP$ multiplication terms must be chosen and combined such that they satisfy correctness constraints on the output matrix. Despite its highly combinatorial nature, this problem has not been exhaustively examined from that perspective, as evidenced for example by the recent deep reinforcement learning approach of AlphaTensor. In this work, we propose a simple yet novel Constraint Programming approach to find non-commutative algorithms for fast matrix multiplication or provide proof of infeasibility otherwise. We propose a set of symmetry-breaking constraints and valid inequalities that are particularly helpful in proving infeasibility. On the feasible side, we find that exploiting solver performance variability in conjunction with a sparsity-based problem decomposition enables finding solutions for larger (feasible) instances of fast matrix multiplication. Our experimental results using CP Optimizer demonstrate that we can find fast matrix multiplication algorithms for matrices up to $3\times 3$ in a short amount of time.
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the distance selection problem is to find the $k$-th smallest interpoint distance among all pairs of points of $P$. The previously best deterministic algorithm solves the problem in $O(n^{4/3} \log^2 n)$ time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to $O(n^{4/3} \log n)$ time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fr\'{e}chet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly $\log^2(m+n)$ (resp., $(m+n)^{\epsilon}$), where $m$ and $n$ are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.
Syntax-guided synthesis is a paradigm in program synthesis in which the search space of candidate solutions is constrained by a syntactic template in the form of a grammar. These syntactic constraints serve two purposes: constraining the language to the space the user desires, but also rendering the search space tractable for the synthesizer. Given a well-written syntactic template, this is an extremely effective technique. However, this is highly dependent on the user providing such a template: a syntactic template that is too large results in a larger search space and slower synthesis, and a syntactic template that is too small may not contain the solution needed. In this work, we frame the space of syntactic templates as a matrix of rules, and demonstrate how this matrix can be searched effectively with little training data using simple search techniques such as genetic algorithms, giving improvements in both the number of benchmarks solved and solving time for the state-of-the-art synthesis solver.
We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha \in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).
The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces, but suffers from computational hardness. The entropic Gromov-Wasserstein (EGW) distance serves as a computationally efficient proxy for the GW distance. Recently, it was shown that the quadratic GW and EGW distances admit variational forms that tie them to the well-understood optimal transport (OT) and entropic OT (EOT) problems. By leveraging this connection, we derive two notions of stability for the EGW problem with the quadratic or inner product cost. The first stability notion enables us to establish convexity and smoothness of the objective in this variational problem. This results in the first efficient algorithms for solving the EGW problem that are subject to formal guarantees in both the convex and non-convex regimes. The second stability notion is used to derive a comprehensive limit distribution theory for the empirical EGW distance and, under additional conditions, asymptotic normality, bootstrap consistency, and semiparametric efficiency thereof.
In recent years, Face Image Quality Assessment (FIQA) has become an indispensable part of the face recognition system to guarantee the stability and reliability of recognition performance in an unconstrained scenario. For this purpose, the FIQA method should consider both the intrinsic property and the recognizability of the face image. Most previous works aim to estimate the sample-wise embedding uncertainty or pair-wise similarity as the quality score, which only considers the information from partial intra-class. However, these methods ignore the valuable information from the inter-class, which is for estimating to the recognizability of face image. In this work, we argue that a high-quality face image should be similar to its intra-class samples and dissimilar to its inter-class samples. Thus, we propose a novel unsupervised FIQA method that incorporates Similarity Distribution Distance for Face Image Quality Assessment (SDD-FIQA). Our method generates quality pseudo-labels by calculating the Wasserstein Distance (WD) between the intra-class similarity distributions and inter-class similarity distributions. With these quality pseudo-labels, we are capable of training a regression network for quality prediction. Extensive experiments on benchmark datasets demonstrate that the proposed SDD-FIQA surpasses the state-of-the-arts by an impressive margin. Meanwhile, our method shows good generalization across different recognition systems.
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