We study the fully dynamic maximum matching problem. In this problem, the goal is to efficiently maintain an approximate maximum matching of a graph that is subject to edge insertions and deletions. Our focus is particularly on algorithms that maintain the edges of a $(1-\epsilon)$-approximate maximum matching for an arbitrarily small constant $\epsilon > 0$. Until recently, the fastest known algorithm for this problem required $\Theta(n)$ time per update where $n$ is the number of vertices. This bound was slightly improved to $n/(\log^* n)^{\Omega(1)}$ by Assadi, Behnezhad, Khanna, and Li [STOC'23] and very recently to $n/2^{\Omega(\sqrt{\log n})}$ by Liu [ArXiv'24]. Whether this can be improved to $n^{1-\Omega(1)}$ remains a major open problem. In this paper, we present a new algorithm that maintains a $(1-\epsilon)$-approximate maximum matching. The update-time of our algorithm is parametrized based on the density of a certain class of graphs that we call Ordered Ruzsa-Szemer\'edi (ORS) graphs, a generalization of the well-known Ruzsa-Szemer\'edi graphs. While determining the density of ORS (or RS) remains a hard problem in combinatorics, we prove that if the existing constructions of ORS graphs are optimal, then our algorithm runs in $n^{1/2+O(\epsilon)}$ time for any fixed $\epsilon > 0$ which would be significantly faster than existing near-linear in $n$ time algorithms. Our second main contribution is a better upper bound on density of both ORS and RS graphs with linear size matchings. The previous best upper bound was due to a proof of the triangle-removal lemma from more than a decade ago due to Fox [Annals of Mathematics '11].
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Dilated convolution, which expands the receptive field by inserting gaps between its consecutive elements, is widely employed in computer vision. In this study, we propose three strategies to improve individual phases of dilated convolution from the view of spectrum analysis. Departing from the conventional practice of fixing a global dilation rate as a hyperparameter, we introduce Frequency-Adaptive Dilated Convolution (FADC), which dynamically adjusts dilation rates spatially based on local frequency components. Subsequently, we design two plug-in modules to directly enhance effective bandwidth and receptive field size. The Adaptive Kernel (AdaKern) module decomposes convolution weights into low-frequency and high-frequency components, dynamically adjusting the ratio between these components on a per-channel basis. By increasing the high-frequency part of convolution weights, AdaKern captures more high-frequency components, thereby improving effective bandwidth. The Frequency Selection (FreqSelect) module optimally balances high- and low-frequency components in feature representations through spatially variant reweighting. It suppresses high frequencies in the background to encourage FADC to learn a larger dilation, thereby increasing the receptive field for an expanded scope. Extensive experiments on segmentation and object detection consistently validate the efficacy of our approach. The code is publicly available at //github.com/Linwei-Chen/FADC.
Nonnegative Matrix Factorization (NMF) is the problem of approximating a given nonnegative matrix M through the conic combination of two nonnegative low-rank matrices W and H. Traditionally NMF is tackled by optimizing a specific objective function evaluating the quality of the approximation. This assessment is often done based on the Frobenius norm. In this study, we argue that the Frobenius norm as the "point-to-point" distance may not always be appropriate. Due to the nonnegative combination resulting in a polyhedral cone, this conic perspective of NMF may not naturally align with conventional point-to-point distance measures. Hence, a ray-to-ray chordal distance is proposed as an alternative way of measuring the discrepancy between M and WH. This measure is related to the Euclidean distance on the unit sphere, motivating us to employ nonsmooth manifold optimization approaches. We apply Riemannian optimization technique to solve chordal-NMF by casting it on a manifold. Unlike existing works on Riemannian optimization that require the manifold to be smooth, the nonnegativity in chordal-NMF is a non-differentiable manifold. We propose a Riemannian Multiplicative Update (RMU) that preserves the convergence properties of Riemannian gradient descent without breaking the smoothness condition on the manifold. We showcase the effectiveness of the Chordal-NMF on synthetic datasets as well as real-world multispectral images.
A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive, implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation. We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to this revenue benchmark when all buyers have unit-demand or additive preferences over m items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.
We perform detailed theoretical analysis of an expectation-maximization-based algorithm recently proposed in for solving a variation of the 3D registration problem, named multi-model 3D registration. Despite having shown superior empirical results, did not theoretically justify the conditions under which the EM approach converges to the ground truth. In this project, we aim to close this gap by establishing such conditions. In particular, the analysis revolves around the usage of probabilistic tail bounds that are developed and applied in various instances throughout the course. The problem studied in this project stands as another example, different from those seen in the course, in which tail-bounds help advance our algorithmic understanding in a probabilistic way. We provide self-contained background materials on 3D Registration
We introduce an algorithm that simplifies the construction of efficient estimators, making them accessible to a broader audience. 'Dimple' takes as input computer code representing a parameter of interest and outputs an efficient estimator. Unlike standard approaches, it does not require users to derive a functional derivative known as the efficient influence function. Dimple avoids this task by applying automatic differentiation to the statistical functional of interest. Doing so requires expressing this functional as a composition of primitives satisfying a novel differentiability condition. Dimple also uses this composition to determine the nuisances it must estimate. In software, primitives can be implemented independently of one another and reused across different estimation problems. We provide a proof-of-concept Python implementation and showcase through examples how it allows users to go from parameter specification to efficient estimation with just a few lines of code.
In a Stackelberg congestion game (SCG), a leader aims to maximize their own gain by anticipating and manipulating the equilibrium state at which the followers settle by playing a congestion game. Often formulated as bilevel programs, large-scale SCGs are well known for their intractability and complexity. Here, we attempt to tackle this computational challenge by marrying traditional methodologies with the latest differentiable programming techniques in machine learning. The core idea centers on replacing the lower-level equilibrium problem with a smooth evolution trajectory defined by the imitative logit dynamic (ILD), which we prove converges to the equilibrium of the congestion game under mild conditions. Building upon this theoretical foundation, we propose two new local search algorithms for SCGs. The first is a gradient descent algorithm that obtains the derivatives by unrolling ILD via differentiable programming. Thanks to the smoothness of ILD, the algorithm promises both efficiency and scalability. The second algorithm adds a heuristic twist by cutting short the followers' evolution trajectory. Behaviorally, this means that, instead of anticipating the followers' best response at equilibrium, the leader seeks to approximate that response by only looking ahead a limited number of steps. Our numerical experiments are carried out over various instances of classic SCG applications, ranging from toy benchmarks to large-scale real-world examples. The results show the proposed algorithms are reliable and scalable local solvers that deliver high-quality solutions with greater regularity and significantly less computational effort compared to the many incumbents included in our study.
We model a Markov decision process, parametrized by an unknown parameter, and study the asymptotic behavior of a sampling-based algorithm, called Thompson sampling. The standard definition of regret is not always suitable to evaluate a policy, especially when the underlying chain structure is general. We show that the standard (expected) regret can grow (super-)linearly and fails to capture the notion of learning in realistic settings with non-trivial state evolution. By decomposing the standard (expected) regret, we develop a new metric, called the expected residual regret, which forgets the immutable consequences of past actions. Instead, it measures regret against the optimal reward moving forward from the current period. We show that the expected residual regret of the Thompson sampling algorithm is upper bounded by a term which converges exponentially fast to 0. We present conditions under which the posterior sampling error of Thompson sampling converges to 0 almost surely. We then introduce the probabilistic version of the expected residual regret and present conditions under which it converges to 0 almost surely. Thus, we provide a viable concept of learning for sampling algorithms which will serve useful in broader settings than had been considered previously.
We consider the problem of distilling efficient network topologies for collective communications. We provide an algorithmic framework for constructing direct-connect topologies optimized for the latency vs. bandwidth trade-off associated with the workload. Our approach synthesizes many different topologies and schedules for a given cluster size and degree and then identifies the appropriate topology and schedule for a given workload. Our algorithms start from small, optimal base topologies and associated communication schedules and use techniques that can be iteratively applied to derive much larger topologies and schedules. Additionally, we incorporate well-studied large-scale graph topologies into our algorithmic framework by producing efficient collective schedules for them using a novel polynomial-time algorithm. Our evaluation uses multiple testbeds and large-scale simulations to demonstrate significant performance benefits from our derived topologies and schedules.
We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate direct solves with a wide variety of symmetric Toeplitz matrices. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
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