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Toward desirable saliency prediction, the types and numbers of inputs for a salient object detection (SOD) algorithm may dynamically change in many real-life applications. However, existing SOD algorithms are mainly designed or trained for one particular type of inputs, failing to be generalized to other types of inputs. Consequentially, more types of SOD algorithms need to be prepared in advance for handling different types of inputs, raising huge hardware and research costs. Differently, in this paper, we propose a new type of SOD task, termed Arbitrary Modality SOD (AM SOD). The most prominent characteristics of AM SOD are that the modality types and modality numbers will be arbitrary or dynamically changed. The former means that the inputs to the AM SOD algorithm may be arbitrary modalities such as RGB, depths, or even any combination of them. While, the latter indicates that the inputs may have arbitrary modality numbers as the input type is changed, e.g. single-modality RGB image, dual-modality RGB-Depth (RGB-D) images or triple-modality RGB-Depth-Thermal (RGB-D-T) images. Accordingly, a preliminary solution to the above challenges, \i.e. a modality switch network (MSN), is proposed in this paper. In particular, a modality switch feature extractor (MSFE) is first designed to extract discriminative features from each modality effectively by introducing some modality indicators, which will generate some weights for modality switching. Subsequently, a dynamic fusion module (DFM) is proposed to adaptively fuse features from a variable number of modalities based on a novel Transformer structure. Finally, a new dataset, named AM-XD, is constructed to facilitate research on AM SOD. Extensive experiments demonstrate that our AM SOD method can effectively cope with changes in the type and number of input modalities for robust salient object detection.

We study robustness to test-time adversarial attacks in the regression setting with $\ell_p$ losses and arbitrary perturbation sets. We address the question of which function classes are PAC learnable in this setting. We show that classes of finite fat-shattering dimension are learnable in both realizable and agnostic settings. Moreover, for convex function classes, they are even properly learnable. In contrast, some non-convex function classes provably require improper learning algorithms. Our main technique is based on a construction of an adversarially robust sample compression scheme of a size determined by the fat-shattering dimension. Along the way, we introduce a novel agnostic sample compression scheme for real-valued functions, which may be of independent interest.

Folklore in complexity theory suspects that circuit lower bounds against $\mathbf{NC}^1$ or $\mathbf{P}/\operatorname{poly}$, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation $\mathbf{NEXP} \not\subseteq \mathbf{P}/\operatorname{poly}$, as recently observed by Pich and Santhanam (2023). We show such a connection conditionally for the Implicit Extended Frege proof system ($\mathsf{iEF}$) introduced by Kraj\'i\v{c}ek (The Journal of Symbolic Logic, 2004), capable of formalizing most of contemporary complexity theory. In particular, we show that if $\mathsf{iEF}$ proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of $\mathsf{iEF}$ proofs implies $\#\mathbf{P} \not\subseteq \mathbf{FP}/\operatorname{poly}$ (which would in turn imply, for example, $\mathbf{PSPACE} \not\subseteq \mathbf{P}/\operatorname{poly}$). Our proof exploits the formalization inside $\mathsf{iEF}$ of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan (Journal of the ACM, 1992). This has consequences for the self-provability of circuit upper bounds in $\mathsf{iEF}$. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

We optimize pipeline parallelism for deep neural network (DNN) inference by partitioning model graphs into $k$ stages and minimizing the running time of the bottleneck stage, including communication. We give practical and effective algorithms for this NP-hard problem, but our emphasis is on tackling the practitioner's dilemma of deciding when a solution is good enough. To this end, we design novel mixed-integer programming (MIP) relaxations for proving lower bounds. Applying these methods to a diverse testbed of 369 production models, for $k \in \{2, 4, 8, 16, 32, 64\}$, we empirically show that these lower bounds are strong enough to be useful in practice. Our lower bounds are substantially stronger than standard combinatorial bounds. For example, evaluated via geometric means across our production testbed with $k = 16$ pipeline stages, our MIP formulations raised the lower bound from 0.4598 to 0.9452, expressed as a fraction of the best partition found. In other words, our improved lower bounds closed the optimality gap by a factor of 9.855x.

If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups.

We study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle $ \Omega \subset \mathbb{R}^3 $. From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on $\Omega$. Then it is possible to follow a single-trace approach to obtain boundary integral equations perturbed by traces of compact volume integral operators with weakly singular kernels. The coupled boundary and volume integral equations are discretized with a Galerkin approach with usual Curl-conforming and Div-conforming finite elements on the boundary and in the volume. Compression techniques and special quadrature rules for singular integrands are required for an efficient and accurate method. Numerical experiments provide evidence that our new formulation enjoys promising properties.

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

We propose new algorithms with provable performance for online binary optimization subject to general constraints and in dynamic settings. We consider the subset of problems in which the objective function is submodular. We propose the online submodular greedy algorithm (OSGA) which solves to optimality an approximation of the previous round loss function to avoid the NP-hardness of the original problem. We extend OSGA to a generic approximation function. We show that OSGA has a dynamic regret bound similar to the tightest bounds in online convex optimization with respect to the time horizon and the cumulative round optimum variation. For instances where no approximation exists or a computationally simpler implementation is desired, we design the online submodular projected gradient descent (OSPGD) by leveraging the Lova\'sz extension. We obtain a regret bound that is akin to the conventional online gradient descent (OGD). Finally, we numerically test our algorithms in two power system applications: fast-timescale demand response and real-time distribution network reconfiguration.

Event-based sensors, distinguished by their high temporal resolution of 1 $\mathrm{\mu}\text{s}$ and a dynamic range of 120 $\text{dB}$, stand out as ideal tools for deployment in fast-paced settings like vehicles and drones. Traditional object detection techniques that utilize Artificial Neural Networks (ANNs) face challenges due to the sparse and asynchronous nature of the events these sensors capture. In contrast, Spiking Neural Networks (SNNs) offer a promising alternative, providing a temporal representation that is inherently aligned with event-based data. This paper explores the unique membrane potential dynamics of SNNs and their ability to modulate sparse events. We introduce an innovative spike-triggered adaptive threshold mechanism designed for stable training. Building on these insights, we present a specialized spiking feature pyramid network (SpikeFPN) optimized for automotive event-based object detection. Comprehensive evaluations demonstrate that SpikeFPN surpasses both traditional SNNs and advanced ANNs enhanced with attention mechanisms. Evidently, SpikeFPN achieves a mean Average Precision (mAP) of 0.477 on the GEN1 Automotive Detection (GAD) benchmark dataset, marking significant increases over the selected SNN baselines. Moreover, the efficient design of SpikeFPN ensures robust performance while optimizing computational resources, attributed to its innate sparse computation capabilities.

The list-labeling problem captures the basic task of storing a dynamically changing set of up to $n$ elements in sorted order in an array of size $m = (1 + \Theta(1))n$. The goal is to support insertions and deletions while moving around elements within the array as little as possible. Until recently, the best known upper bound stood at $O(\log^2 n)$ amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized $O(\log^{3/2} n)$ expected-cost algorithm was discovered. The best randomized lower bound for this problem remains $\Omega(\log n)$, and closing this gap is considered to be a major open problem in data structures. In this paper, we present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of $O(\log n \operatorname{polyloglog} n)$ amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions.