In this paper, we put forth a novel framework (named ``RYU'') for the construction of ``safe'' balls, i.e. regions that provably contain the dual solution of a target optimization problem. We concentrate on the standard setup where the cost function is the sum of two terms: a closed, proper, convex Lipschitz-smooth function and a closed, proper, convex function. The RYU framework is shown to generalize or improve upon all the results proposed in the last decade for the considered family of optimization problems.
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In this paper, we introduce CDL, a software library designed for the analysis of permutations and linear orders subject to various structural restrictions. Prominent examples of these restrictions include pattern avoidance, a topic of interest in both computer science and combinatorics, and "never conditions" utilized in social choice and voting theory. CDL offers a range of fundamental functionalities, including identifying the permutations that meet specific restrictions and determining the isomorphism of such sets. To facilitate exploration of large permutation sets or domains, CDL incorporates multiple search strategies and heuristics.
In this paper, we present a novel deep image clustering approach termed PICI, which enforces the partial information discrimination and the cross-level interaction in a joint learning framework. In particular, we leverage a Transformer encoder as the backbone, through which the masked image modeling with two paralleled augmented views is formulated. After deriving the class tokens from the masked images by the Transformer encoder, three partial information learning modules are further incorporated, including the PISD module for training the auto-encoder via masked image reconstruction, the PICD module for employing two levels of contrastive learning, and the CLI module for mutual interaction between the instance-level and cluster-level subspaces. Extensive experiments have been conducted on six real-world image datasets, which demononstrate the superior clustering performance of the proposed PICI approach over the state-of-the-art deep clustering approaches. The source code is available at //github.com/Regan-Zhang/PICI.
In this paper, we provide conditions that hulls of generalized Reed-Solomon (GRS) codes are also GRS codes from algebraic geometry codes. If the conditions are not satisfied, we provide a method of linear algebra to find the bases of hulls of GRS codes and give formulas to compute their dimensions. Besides, we explain that the conditions are too good to be improved by some examples. Moreover, we show self-orthogonal and self-dual GRS codes.
In this note we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank-Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed. In this note we study for the temporal discretization an interesting family of diagonally implicit Runge-Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter $\theta$ is preferable.
In this paper, we make the explicit connection between image segmentation methods and end-to-end diarization methods. From these insights, we propose a novel, fully end-to-end diarization model, EEND-M2F, based on the Mask2Former architecture. Speaker representations are computed in parallel using a stack of transformer decoders, in which irrelevant frames are explicitly masked from the cross attention using predictions from previous layers. EEND-M2F is lightweight, efficient, and truly end-to-end, as it does not require any additional diarization, speaker verification, or segmentation models to run, nor does it require running any clustering algorithms. Our model achieves state-of-the-art performance on several public datasets, such as AMI, AliMeeting and RAMC. Most notably our DER of 16.07% on DIHARD-III is the first major improvement upon the challenge winning system.
In this paper, we present refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube with the Neumann boundary condition. And sharper generalization bounds are derived based on the localization techniques under the assumptions that the exact solutions of the PDEs lie in the Barron space or the general Sobolev spaces. For the PINNs, we investigate the general linear second elliptic PDEs with Dirichlet boundary condition.
We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.
This paper presents NOMAD (Non-Matching Audio Distance), a differentiable perceptual similarity metric that measures the distance of a degraded signal against non-matching references. The proposed method is based on learning deep feature embeddings via a triplet loss guided by the Neurogram Similarity Index Measure (NSIM) to capture degradation intensity. During inference, the similarity score between any two audio samples is computed through Euclidean distance of their embeddings. NOMAD is fully unsupervised and can be used in general perceptual audio tasks for audio analysis e.g. quality assessment and generative tasks such as speech enhancement and speech synthesis. The proposed method is evaluated with 3 tasks. Ranking degradation intensity, predicting speech quality, and as a loss function for speech enhancement. Results indicate NOMAD outperforms other non-matching reference approaches in both ranking degradation intensity and quality assessment, exhibiting competitive performance with full-reference audio metrics. NOMAD demonstrates a promising technique that mimics human capabilities in assessing audio quality with non-matching references to learn perceptual embeddings without the need for human-generated labels.
Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of the marginal likelihood. The learned harmonic mean estimator learns an importance sampling target distribution that approximates the optimal distribution. While the approximation need not be highly accurate, it is critical that the probability mass of the learned distribution is contained within the posterior in order to avoid the exploding variance problem. In previous work a bespoke optimization problem is introduced when training models in order to ensure this property is satisfied. In the current article we introduce the use of normalizing flows to represent the importance sampling target distribution. A flow-based model is trained on samples from the posterior by maximum likelihood estimation. Then, the probability density of the flow is concentrated by lowering the variance of the base distribution, i.e. by lowering its "temperature", ensuring its probability mass is contained within the posterior. This approach avoids the need for a bespoke optimisation problem and careful fine tuning of parameters, resulting in a more robust method. Moreover, the use of normalizing flows has the potential to scale to high dimensional settings. We present preliminary experiments demonstrating the effectiveness of the use of flows for the learned harmonic mean estimator. The harmonic code implementing the learned harmonic mean, which is publicly available, has been updated to now support normalizing flows.
In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in $f(k)n^{O(1)}$ time and $f(k)\log n$ space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on `tree-structured graphs' are complete for this class: we show that List Colouring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by $\log n$, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a `natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most $f(k)n^{O(1)}$ and use $f(k)\log n$ space. Moreover, we introduce `tree-shaped' variants of Weighted CNF-Satisfiability and Multicolour Clique that are XALP-complete.
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