We address the challenges in estimating 3D human poses from multiple views under occlusion and with limited overlapping views. We approach multi-view, single-person 3D human pose reconstruction as a regression problem and propose a novel encoder-decoder Transformer architecture to estimate 3D poses from multi-view 2D pose sequences. The encoder refines 2D skeleton joints detected across different views and times, fusing multi-view and temporal information through global self-attention. We enhance the encoder by incorporating a geometry-biased attention mechanism, effectively leveraging geometric relationships between views. Additionally, we use detection scores provided by the 2D pose detector to further guide the encoder's attention based on the reliability of the 2D detections. The decoder subsequently regresses the 3D pose sequence from these refined tokens, using pre-defined queries for each joint. To enhance the generalization of our method to unseen scenes and improve resilience to missing joints, we implement strategies including scene centering, synthetic views, and token dropout. We conduct extensive experiments on three benchmark public datasets, Human3.6M, CMU Panoptic and Occlusion-Persons. Our results demonstrate the efficacy of our approach, particularly in occluded scenes and when few views are available, which are traditionally challenging scenarios for triangulation-based methods.

Accurate extrinsic calibration of wide baseline multi-camera systems enables better understanding of 3D scenes for many applications and is of great practical importance. Classical Structure-from-Motion calibration methods require special calibration equipment so that accurate point correspondences can be detected between different views. In addition, an operator with some training is usually needed to ensure that data is collected in a way that leads to good calibration accuracy. This limits the ease of adoption of such technologies. Recently, methods have been proposed to use human pose estimation models to establish point correspondences, thus removing the need for any special equipment. The challenge with this approach is that human pose estimation algorithms typically produce much less accurate feature points compared to classical patch-based methods. Another problem is that ambient human motion might not be optimal for calibration. We build upon prior works and introduce several novel ideas to improve the accuracy of human-pose-based extrinsic calibration. Our first contribution is a robust reprojection loss based on a better understanding of the sources of pose estimation error. Our second contribution is a 3D human pose likelihood model learned from motion capture data. We demonstrate significant improvements in calibration accuracy by evaluating our method on four publicly available datasets.

We study $k$-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least $(1 - δ)$, simultaneously has a cost of at most $(1 + ε)$ times the optimal cost and an accuracy of at least $(1 - ε)$? We show how to achieve such a clustering on $n$ points with $O{((k^2 \log n) \cdot m{(Q, ε^4, δ/ (k\log n))})}$ oracle queries, when the $k$ clusters can be learned with an $ε'$ error and a failure probability $δ'$ using $m(Q, ε',δ')$ labeled samples in the supervised setting, where $Q$ is the set of candidate cluster centers. We show that $m(Q, ε', δ')$ is small both for $k$-means instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean $k$-means instances, we can avoid the dependency on $n$ in the query complexity at the expense of an increased dependency on $k$: specifically, we give a slightly more involved algorithm that uses $O(k^4/(ε^2 δ) + (k^{9}/ε^4) \log(1/δ) + k \cdot m(\mathbb{R}^r, ε^4/k, δ))$ oracle queries. We also show that the number of queries needed for $(1 - ε)$-accuracy in Euclidean $k$-means must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space $k$-means, we show that it must at least be logarithmic in the number of candidate centers. This shows that our query complexities capture the right dependencies on the respective parameters.

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any ∊ > 0, there is a (1/∊)O(1)nO(log n)-size LP relaxation with an integrality gap of at most 2 + ∊, where n is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on O (log2 n)-depth monotone circuits with fan-in 2 for evaluating weighted threshold functions with n inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.

An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it's possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.

We prove that for all integer k ≥ 3, there is a predicate P on k Boolean variables with 2~O(k1/3) accepting assignments that is approximation resistant even on satisfiable instances. That is, given a satisfiable CSP instance with constraint P, we cannot achieve better approximation ratio than simply picking random assignments. This improves the best previously known result by Hastad and Khot where the predicate has 2O(k1/2) accepting assignments. Our construction is inspired by several recent developments. One is the idea of using direct sums to improve soundness of PCPs, developed by Chan [5]. We also use techniques from Wenner [32] to construct PCPs with perfect completeness without relying on the d-to-1 Conjecture.

Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as #CSP and Graph Homomorphism. An effective dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and Graph Homomorphism can be viewed as sub-families of Holant with the additional assumption that the equality constraints are always available. Other sub-families of Holant such as Holant* and Holantcproblems, in which we assume some specific sets of constraints to be freely available, were also studied. The Holant framework becomes more expressive and contains more interesting tractable cases with less or no freely available constraint functions, while, on the other hand, it also becomes more challenging to obtain a complete characterization of its time complexity. Recently, complexity dichotomy for a variety of sub-families of Holant such as #CSP, Graph Homomorphism, Holant* and Holantcwere proved. The dichotomy for the general Holant framework, which is the most desirable, still remains open. In this paper, we prove a dichotomy for the general Holant framework where all the constraints are real symmetric functions. This setting already captures most of the interesting combinatorial problems defined by local constraints, such as (perfect) matching, independent set, vertex cover and so on. This is the first time a dichotomy is obtained for general Holant Problems without any auxiliary functions. One benefit of working with Holant framework is some powerful new reduction techniques such as Holographic reduction. Along the proof of our dichotomy, we introduce a new reduction technique, namely realizing a constraint function by approximating it. This new technique is employed in our proof in a situation where it seems that all previous reduction techniques fail, thus this new idea of reduction might also be of independent interest. Besides proving dichotomy and developing new technique, we also obtained some interesting by-products. We prove a dichotomy for #CSP restricting to instances where each variable appears a multiple of d times for any d. We also prove that counting the number of Eulerian-Orientations on 2k-regular graphs is #P-hard for any k≥2.

We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k for any positive integer $k>1$. This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k.