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We give a $\widetilde{O}(n)$ time sampler for independent sets of a matroid with $n$ elements. As an application, there is a near-linear time sampler for the all-terminal network reliability.

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.

Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query object $\gamma$, which is also a plate, we can quickly answer various intersection queries, such as detecting whether $\gamma$ intersects any plate of $\mathcal{T}$, reporting all the plates intersected by $\gamma$, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree parametrized algebraic arcs in $\mathbb{R}^3$ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in $\mathbb{R}^3$. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we present many different data structures for intersection queries, which also provide trade-offs between their size and query time. For example, if $\mathcal{T}$ is a set of plates and the query objects are algebraic arcs, we obtain a data structure that uses $O^*(n^{4/3})$ storage (where the $O^*(\cdot)$ notation hides subpolynomial factors) and answers an arc-intersection query in $O^*(n^{2/3})$ time. This result is significant since the exponents do not depend on the specific shape of the input and query objects. For a parameter $s\in [n^{4/3}, n^{t_Q}]$ where $t_Q\ge 3$ is the number of real parameters needed to specify a query arc, the query time can be decreased to $O^*((n/s^{1/t_Q})^{\tfrac{2}{3}(1-1/t_Q)})$ by increasing the storage to $O^*(s)$.

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.

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.

We give an algorithm that takes as input an $n$-vertex graph $G$ and an integer $k$, runs in time $2^{O(k^2)} n^{O(1)}$, and outputs a tree decomposition of $G$ of width at most $k$, if such a decomposition exists. This resolves the long-standing open problem of whether there is a $2^{o(k^3)} n^{O(1)}$ time algorithm for treewidth. In particular, our algorithm is the first improvement on the dependency on $k$ in algorithms for treewidth since the $2^{O(k^3)} n^{O(1)}$ time algorithm given by Bodlaender and Kloks [ICALP 1991] and Lagergren and Arnborg [ICALP 1991]. We also give an algorithm that given an $n$-vertex graph $G$, an integer $k$, and a rational $\varepsilon \in (0,1)$, in time $k^{O(k/\varepsilon)} n^{O(1)}$ either outputs a tree decomposition of $G$ of width at most $(1+\varepsilon)k$ or determines that the treewidth of $G$ is larger than $k$. Prior to our work, no approximation algorithms for treewidth with approximation ratio less than $2$, other than the exact algorithms, were known. Both of our algorithms work in polynomial space.

Chain-of-Though (CoT) prompting has shown promising performance in various reasoning tasks. Recently, Self-Consistency \citep{wang2023selfconsistency} proposes to sample a diverse set of reasoning chains which may lead to different answers while the answer that receives the most votes is selected. In this paper, we propose a novel method to use backward reasoning in verifying candidate answers. We mask a token in the question by ${\bf x}$ and ask the LLM to predict the masked token when a candidate answer is provided by \textit{a simple template}, i.e., ``\textit{\textbf{If we know the answer of the above question is \{a candidate answer\}, what is the value of unknown variable ${\bf x}$?}}'' Intuitively, the LLM is expected to predict the masked token successfully if the provided candidate answer is correct. We further propose FOBAR to combine forward and backward reasoning for estimating the probability of candidate answers. We conduct extensive experiments on six data sets and three LLMs. Experimental results demonstrate that FOBAR achieves state-of-the-art performance on various reasoning benchmarks.

Whilst a majority of affective computing research focuses on inferring emotions, examining mood or understanding the \textit{mood-emotion interplay} has received significantly less attention. Building on prior work, we (a) deduce and incorporate emotion-change ($\Delta$) information for inferring mood, without resorting to annotated labels, and (b) attempt mood prediction for long duration video clips, in alignment with the characterisation of mood. We generate the emotion-change ($\Delta$) labels via metric learning from a pre-trained Siamese Network, and use these in addition to mood labels for mood classification. Experiments evaluating \textit{unimodal} (training only using mood labels) vs \textit{multimodal} (training using mood plus $\Delta$ labels) models show that mood prediction benefits from the incorporation of emotion-change information, emphasising the importance of modelling the mood-emotion interplay for effective mood inference.

We give a quantum approximation scheme (i.e., $(1 + \varepsilon)$-approximation for every $\varepsilon > 0$) for the classical $k$-means clustering problem in the QRAM model with a running time that has only polylogarithmic dependence on the number of data points. More specifically, given a dataset $V$ with $N$ points in $\mathbb{R}^d$ stored in QRAM data structure, our quantum algorithm runs in time $\tilde{O} \left( 2^{\tilde{O}(\frac{k}{\varepsilon})} \eta^2 d\right)$ and with high probability outputs a set $C$ of $k$ centers such that $cost(V, C) \leq (1+\varepsilon) \cdot cost(V, C_{OPT})$. Here $C_{OPT}$ denotes the optimal $k$-centers, $cost(.)$ denotes the standard $k$-means cost function (i.e., the sum of the squared distance of points to the closest center), and $\eta$ is the aspect ratio (i.e., the ratio of maximum distance to minimum distance). This is the first quantum algorithm with a polylogarithmic running time that gives a provable approximation guarantee of $(1+\varepsilon)$ for the $k$-means problem. Also, unlike previous works on unsupervised learning, our quantum algorithm does not require quantum linear algebra subroutines and has a running time independent of parameters (e.g., condition number) that appear in such procedures.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.