A general, {\em rectangular} kernel matrix may be defined as $K_{ij} = \kappa(x_i,y_j)$ where $\kappa(x,y)$ is a kernel function and where $X=\{x_i\}_{i=1}^m$ and $Y=\{y_i\}_{i=1}^n$ are two sets of points. In this paper, we seek a low-rank approximation to a kernel matrix where the sets of points $X$ and $Y$ are large and are arbitrarily distributed, such as away from each other, ``intermingled'', identical, etc. Such rectangular kernel matrices may arise, for example, in Gaussian process regression where $X$ corresponds to the training data and $Y$ corresponds to the test data. In this case, the points are often high-dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly or nearly linear with respect to the size of data for a fixed approximation rank. The main idea in this paper is to {\em geometrically} select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed.
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Out-of-distribution (OOD) detection refers to training the model on an in-distribution (ID) dataset to classify whether the input images come from unknown classes. Considerable effort has been invested in designing various OOD detection methods based on either convolutional neural networks or transformers. However, zero-shot OOD detection methods driven by CLIP, which only require class names for ID, have received less attention. This paper presents a novel method, namely CLIP saying "no" (\textbf{CLIPN}), which empowers the logic of saying "no" within CLIP. Our key motivation is to equip CLIP with the capability of distinguishing OOD and ID samples using positive-semantic prompts and negation-semantic prompts. Specifically, we design a novel learnable "no" prompt and a "no" text encoder to capture negation semantics within images. Subsequently, we introduce two loss functions: the image-text binary-opposite loss and the text semantic-opposite loss, which we use to teach CLIPN to associate images with "no" prompts, thereby enabling it to identify unknown samples. Furthermore, we propose two threshold-free inference algorithms to perform OOD detection by utilizing negation semantics from "no" prompts and the text encoder. Experimental results on 9 benchmark datasets (3 ID datasets and 6 OOD datasets) for the OOD detection task demonstrate that CLIPN, based on ViT-B-16, outperforms 7 well-used algorithms by at least 2.34\% and 11.64\% in terms of AUROC and FPR95 for zero-shot OOD detection on ImageNet-1K. Our CLIPN can serve as a solid foundation for effectively leveraging CLIP in downstream OOD tasks. The code is available on //github.com/xmed-lab/CLIPN}{//github.com/xmed-lab/CLIPN.
The Ramsey number is the minimum number of nodes, $n = R(s, t)$, such that all undirected simple graphs of order $n$, contain a clique of order $s$, or an independent set of order $t$. This paper explores the application of a best first search algorithm and reinforcement learning (RL) techniques to find counterexamples to specific Ramsey numbers. We incrementally improve over prior search methods such as random search by introducing a graph vectorization and deep neural network (DNN)-based heuristic, which gauge the likelihood of a graph being a counterexample. The paper also proposes algorithmic optimizations to confine a polynomial search runtime. This paper does not aim to present new counterexamples but rather introduces and evaluates a framework supporting Ramsey counterexample exploration using other heuristics. Code and methods are made available through a PyPI package and GitHub repository.
We study the equivalence testing problem where the goal is to determine if the given two unknown distributions on $[n]$ are equal or $\epsilon$-far in the total variation distance in the conditional sampling model (CFGM, SICOMP16; CRS, SICOMP15) wherein a tester can get a sample from the distribution conditioned on any subset. Equivalence testing is a central problem in distribution testing, and there has been a plethora of work on this topic in various sampling models. Despite significant efforts over the years, there remains a gap in the current best-known upper bound of $\tilde{O}(\log \log n)$ [FJOPS, COLT 2015] and lower bound of $\Omega(\sqrt{\log \log n})$[ACK, RANDOM 2015, Theory of Computing 2018]. Closing this gap has been repeatedly posed as an open problem (listed as problems 66 and 87 at sublinear.info). In this paper, we completely resolve the query complexity of this problem by showing a lower bound of $\tilde{\Omega}(\log \log n)$. For that purpose, we develop a novel and generic proof technique that enables us to break the $\sqrt{\log \log n}$ barrier, not only for the equivalence testing problem but also for other distribution testing problems, such as uniblock property.
We explore element-wise convex combinations of two permutation-aligned neural network parameter vectors $\Theta_A$ and $\Theta_B$ of size $d$. We conduct extensive experiments by examining various distributions of such model combinations parametrized by elements of the hypercube $[0,1]^{d}$ and its vicinity. Our findings reveal that broad regions of the hypercube form surfaces of low loss values, indicating that the notion of linear mode connectivity extends to a more general phenomenon which we call mode combinability. We also make several novel observations regarding linear mode connectivity and model re-basin. We demonstrate a transitivity property: two models re-based to a common third model are also linear mode connected, and a robustness property: even with significant perturbations of the neuron matchings the resulting combinations continue to form a working model. Moreover, we analyze the functional and weight similarity of model combinations and show that such combinations are non-vacuous in the sense that there are significant functional differences between the resulting models.
The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = \omega(1)$, and $n = \tilde \omega(d^{5/4})$. We conjecture that $n = \omega(d \log d)$ suffices for the same assertion to hold. The factor $d^{-1/2}$ normalising $R$ is essentially best possible.
We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to express that a vertex v is contained in a hyperedge e. We show that two hypergraphs H, H' satisfy the same sentences of the logic $GC^k$ if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of generalised hypertree width at most k. Here, H, H' are called homomorphism indistinguishable over a class C if for every hypergraph G in C the number of homomorphisms from G to H equals the number of homomorphisms from G to H'. This result can be viewed as a generalisation (from graphs to hypergraphs) of a result by Dvorak (2010) stating that any two (undirected, simple, finite) graphs H, H' are indistinguishable by the (k+1)-variable counting logic $C^{k+1}$ if, and only if, they are homomorphism indistinguishable on the class of graphs of tree width at most k.
Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries specified by a set of entries $\Omega\subseteq [m]\times [n]$. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~\cite{jns13} showed that if $M$ has incoherent rows and columns, then alternating minimization provably recovers the matrix $M$ by observing a nearly linear in $n$ number of entries. While the sample complexity has been subsequently improved~\cite{glz17}, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time $\widetilde O(|\Omega| k)$, which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require $\widetilde O(|\Omega| k^2)$ time.
We consider an inverse problem $\mathbf{y}= f(\mathbf{Ax})$, where $\mathbf{x}\in\mathbb{R}^n$ is the signal of interest, $\mathbf{A}$ is the sensing matrix, $f$ is a nonlinear function and $\mathbf{y} \in \mathbb{R}^m$ is the measurement vector. In many applications, we have some level of freedom to design the sensing matrix $\mathbf{A}$, and in such circumstances we could optimize $\mathbf{A}$ to achieve better reconstruction performance. As a first step towards optimal design, it is important to understand the impact of the sensing matrix on the difficulty of recovering $\mathbf{x}$ from $\mathbf{y}$. In this paper, we study the performance of one of the most successful recovery methods, i.e., the expectation propagation (EP) algorithm. We define a notion of spikiness for the spectrum of $\bmmathbfA}$ and show the importance of this measure for the performance of EP. We show that whether a spikier spectrum can hurt or help the recovery performance depends on $f$. Based on our framework, we are able to show that, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky spectrums lead to better performance. Our results unify and substantially generalize existing results that compare Gaussian and orthogonal matrices, and provide a platform towards designing optimal sensing systems.
A $\mu$-constrained Boolean Max-CSP$(\psi)$ instance is a Boolean Max-CSP instance on predicate $\psi:\{0,1\}^r \to \{0,1\}$ where the objective is to find a labeling of relative weight exactly $\mu$ that maximizes the fraction of satisfied constraints. In this work, we study the approximability of constrained Boolean Max-CSPs via SDP hierarchies by relating the integrality gap of Max-CSP $(\psi)$ to its $\mu$-dependent approximation curve. Formally, assuming the Small-Set Expansion Hypothesis, we show that it is NP-hard to approximate $\mu$-constrained instances of Max-CSP($\psi$) up to factor ${\sf Gap}_{\ell,\mu}(\psi)/\log(1/\mu)^2$ (ignoring factors depending on $r$) for any $\ell \geq \ell(\mu,r)$. Here, ${\sf Gap}_{\ell,\mu}(\psi)$ is the optimal integrality gap of $\ell$-round Lasserre relaxation for $\mu$-constrained Max-CSP($\psi$) instances. Our results are derived by combining the framework of Raghavendra [STOC 2008] along with more recent advances in rounding Lasserre relaxations and reductions from the Small-Set Expansion (SSE) problem. A crucial component of our reduction is a novel way of composing generic bias-dependent dictatorship tests with SSE, which could be of independent interest.
Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions ($n=\infty$) the problem is solved optimally by the likelihood-ratio test; when $m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-off between $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulation data needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.
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