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Novelty detection is a fundamental task of machine learning which aims to detect abnormal ($\textit{i.e.}$ out-of-distribution (OOD)) samples. Since diffusion models have recently emerged as the de facto standard generative framework with surprising generation results, novelty detection via diffusion models has also gained much attention. Recent methods have mainly utilized the reconstruction property of in-distribution samples. However, they often suffer from detecting OOD samples that share similar background information to the in-distribution data. Based on our observation that diffusion models can \emph{project} any sample to an in-distribution sample with similar background information, we propose \emph{Projection Regret (PR)}, an efficient novelty detection method that mitigates the bias of non-semantic information. To be specific, PR computes the perceptual distance between the test image and its diffusion-based projection to detect abnormality. Since the perceptual distance often fails to capture semantic changes when the background information is dominant, we cancel out the background bias by comparing it against recursive projections. Extensive experiments demonstrate that PR outperforms the prior art of generative-model-based novelty detection methods by a significant margin.

In the literature on algorithms for performing the multi-term addition $s_n=\sum_{i=1}^n x_i$ using floating-point arithmetic it is often shown that a hardware unit that has single normalization and rounding improves precision, area, latency, and power consumption, compared with the use of standard add or fused multiply-add units. However, non-monotonicity can appear when computing sums with a subclass of multi-term addition units, which currently is not explored in the literature. We demonstrate that common techniques for performing multi-term addition with $n\geq 4$, without normalization of intermediate quantities, can result in non-monotonicity -- increasing one of the addends $x_i$ decreases the sum $s_n$. Summation is required in dot product and matrix multiplication operations, operations that have increasingly started appearing in the hardware of supercomputers, thus knowing where monotonicity is preserved can be of interest to the users of these machines. Our results suggest that non-monotonicity of summation, in some of the commercial hardware devices that implement a specific class of multi-term adders, is a feature that may have appeared unintentionally as a consequence of design choices that reduce circuit area and other metrics. To demonstrate our findings, we use formal proofs as well as a numerical simulation of non-monotonic multi-term adders in MATLAB.

We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. In this paper, we study the approximability of the problem on a large variety of graph classes. Our first result is a linear-time $(1+\varepsilon)$-approximation reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This directly yields linear-size $(1+\varepsilon)$-approximation lossy kernels for the problems on any bounded-expansion graph classes. Our second result is a linear-time approximation scheme for (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of polynomial expansion, based on the local-search framework of Har-Peled and Quanrud [SICOMP 2017]. This approximation scheme can be applied to a more general family of problems that aim to hit all subgraphs satisfying a certain property $\pi$ that is efficiently testable and has bounded diameter. Both of our results have applications to Subgraph Hitting (not induced) on wide classes of geometric intersection graphs, resulting in linear-size lossy kernels and (near-)linear time approximation schemes for the problem.

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)} n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach can be generalized to the problem with $\ell_p$ metric for any rational $1 \le p \le \infty$, or even other metrics on $\mathbb{R}^2$.

Consider that there are $k\le n$ agents in a simple, connected, and undirected graph $G=(V,E)$ with $n$ nodes and $m$ edges. The goal of the dispersion problem is to move these $k$ agents to distinct nodes. Agents can communicate only when they are at the same node, and no other means of communication such as whiteboards are available. We assume that the agents operate synchronously. We consider two scenarios: when all agents are initially located at any single node (rooted setting) and when they are initially distributed over any one or more nodes (general setting). Kshemkalyani and Sharma presented a dispersion algorithm for the general setting, which uses $O(m_k)$ time and $\log(k+\delta)$ bits of memory per agent [OPODIS 2021]. Here, $m_k$ is the maximum number of edges in any induced subgraph of $G$ with $k$ nodes, and $\delta$ is the maximum degree of $G$. This algorithm is the fastest in the literature, as no algorithm with $o(m_k)$ time has been discovered even for the rooted setting. In this paper, we present faster algorithms for both the rooted and general settings. First, we present an algorithm for the rooted setting that solves the dispersion problem in $O(k\log \min(k,\delta))=O(k\log k)$ time using $O(\log \delta)$ bits of memory per agent. Next, we propose an algorithm for the general setting that achieves dispersion in $O(k (\log k)\cdot (\log \min(k,\delta))=O(k \log^2 k)$ time using $O(\log (k+\delta))$ bits. Finally, for the rooted setting, we give a time-optimal, i.e.,$O(k)$-time, algorithm with $O(\delta)$ bits of space per agent.

The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ and called the standardly extended code of $\C$, is a linear code with parameters $[n+1, k, \bar{d}]$, where $\bar{d}=d$ or $\bar{d}=d+1$. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.

Non-autoregressive models have been widely studied in the Complete Information Scenario (CIS), in which the input has complete information of corresponding output. However, their explorations in the Incomplete Information Scenario (IIS) are extremely limited. Our analyses reveal that the IIS's incomplete input information will augment the inherent limitations of existing non-autoregressive models trained under Maximum Likelihood Estimation. In this paper, we propose for the IIS an Adversarial Non-autoregressive Transformer (ANT) which has two features: 1) Position-Aware Self-Modulation to provide more reasonable hidden representations, and 2) Dependency Feed Forward Network to strengthen its capacity in dependency modeling. We compare ANT with other mainstream models in the IIS and demonstrate that ANT can achieve comparable performance with much fewer decoding iterations. Furthermore, we show its great potential in various applications like latent interpolation and semi-supervised learning.

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $p=1$ or $p=\infty$. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be unbounded continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f) \cap\mathcal{D}(\theta_g))\setminus\{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad \mu(\operatorname{supp}(\theta_f x))\nu(\operatorname{supp}(\theta_g x)) \geq \frac{1}{\left(\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|\right)\left(\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|\right)}, \end{align} where \begin{align*} &\theta_f:\mathcal{D}(\theta_f) \ni x \mapsto \theta_fx \in \mathcal{L}^p(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{D}(\theta_g) \ni x \mapsto \theta_gx \in \mathcal{L}^p(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} We call Inequality (1) as \textbf{Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Along with recent \textbf{Functional Continuous Uncertainty Principle} [arXiv:2308.00312], Inequality (1) also improves Ricaud-Torr\'{e}sani uncertainty principle [IEEE Trans. Inform. Theory, 2013]. In particular, it improves Elad-Bruckstein uncertainty principle [IEEE Trans. Inform. Theory, 2002] and Donoho-Stark uncertainty principle [SIAM J. Appl. Math., 1989].

In the matroid partitioning problem, we are given $k$ matroids $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$ defined over a common ground set $V$ of $n$ elements, and we need to find a partitionable set $S \subseteq V$ of largest possible cardinality, denoted by $p$. Here, a set $S \subseteq V$ is called partitionable if there exists a partition $(S_1, \dots , S_k)$ of $S$ with $S_i \in \mathcal{I}_i$ for $i = 1, \ldots, k$. In 1986, Cunningham [SICOMP 1986] presented a matroid partition algorithm that uses $O(n p^{3/2} + k n)$ independence oracle queries, which was the previously known best algorithm. This query complexity is $O(n^{5/2})$ when $k \leq n$. Our main result is to present a matroid partition algorithm that uses $\tilde{O}(k'^{1/3} n p + k n)$ independence oracle queries, where $k' = \min\{k, p\}$. This query complexity is $\tilde{O}(n^{7/3})$ when $k \leq n$, and this improves upon the one of previous Cunningham's algorithm. To obtain this, we present a new approach \emph{edge recycling augmentation}, which can be attained through new ideas: an efficient utilization of the binary search technique by Nguyen [2019] and Chakrabarty-Lee-Sidford-Singla-Wong [FOCS 2019] and a careful analysis of the independence oracle query complexity. Our analysis differs significantly from the one for matroid intersection algorithms, because of the parameter $k$. We also present a matroid partition algorithm that uses $\tilde{O}((n + k) \sqrt{p})$ rank oracle queries.

We consider the optimization problem of cardinality constrained maximization of a monotone submodular set function $f:2^U\to\mathbb{R}_{\geq 0}$ (SM) with noisy evaluations of $f$. In particular, it is assumed that we do not have value oracle access to $f$, but instead for any $X\subseteq U$ and $u\in U$ we can take samples from a noisy distribution with expected value $f(X\cup\{u\})-f(X)$. Our goal is to develop algorithms in this setting that take as few samples as possible, and return a solution with an approximation guarantee relative to the optimal with high probability. We propose the algorithm Confident Threshold Greedy (CTG), which is based on the threshold greedy algorithm of Badanidiyuru and Vondrak [1] and samples adaptively in order to produce an approximate solution with high probability. We prove that CTG achieves an approximation ratio arbitrarily close to $1-1/e$, depending on input parameters. We provide an experimental evaluation on real instances of SM and demonstrate the sample efficiency of CTG.