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Mahalanobis metrics are widely used in machine learning in conjunction with methods like $k$-nearest neighbors, $k$-means clustering, and $k$-medians clustering. Despite their importance, there has not been any prior work on applying sketching techniques to speed up algorithms for Mahalanobis metrics. In this paper, we initiate the study of dimension reduction for Mahalanobis metrics. In particular, we provide efficient data structures for solving the Approximate Distance Estimation (ADE) problem for Mahalanobis distances. We first provide a randomized Monte Carlo data structure. Then, we show how we can adapt it to provide our main data structure which can handle sequences of \textit{adaptive} queries and also online updates to both the Mahalanobis metric matrix and the data points, making it amenable to be used in conjunction with prior algorithms for online learning of Mahalanobis metrics.

We study the influence minimization problem: given a graph $G$ and a seed set $S$, blocking at most $b$ nodes or $b$ edges such that the influence spread of the seed set is minimized. This is a pivotal yet underexplored aspect of network analytics, which can limit the spread of undesirable phenomena in networks, such as misinformation and epidemics. Given the inherent NP-hardness of the problem under the IC and LT models, previous studies have employed greedy algorithms and Monte Carlo Simulations for its resolution. However, existing techniques become cost-prohibitive when applied to large networks due to the necessity of enumerating all the candidate blockers and computing the decrease in expected spread from blocking each of them. This significantly restricts the practicality and effectiveness of existing methods, especially when prompt decision-making is crucial. In this paper, we propose the AdvancedGreedy algorithm, which utilizes a novel graph sampling technique that incorporates the dominator tree structure. We find that AdvancedGreedy can achieve a $(1-1/e-\epsilon)$-approximation in the problem under the LT model. Experimental evaluations on real-life networks reveal that our proposed algorithms exhibit a significant enhancement in efficiency, surpassing the state-of-the-art algorithm by three orders of magnitude, while achieving high effectiveness.

Nash equilibrium} (NE) can be stated as a formal theorem on a multilinear form, free of game theory terminology. On the other hand, inspired by this formalism, we state and prove a {\it multilinear minimax theorem}, a generalization of von Neumann's bilinear minimax theorem. As in the bilinear case, the proof is based on relating the underlying optimizations to a primal-dual pair of linear programming problems, albeit more complicated LPs. The theorem together with its proof is of independent interest. Next, we use the theorem to associate to a multilinear form in NE a {\it multilinear minimax relaxation} (MMR), where the primal-dual pair of solutions induce an approximate equilibrium point that provides a nontrivial upper bound on a convex combination of {\it expected payoffs} in any NE solution. In fact we show any positive probability vector associated to the players induces a corresponding {\it diagonally-scaled} MMR approximate equilibrium with its associated upper bound. By virtue of the proof of the multilinear minimax theorem, MMR solution can be computed in polynomial-time. On the other hand, it is known that even in bimatrix games NE is {\it PPAD-complete}, a complexity class in NP not known to be in P. The quality of MMR solution and the efficiency of solving the underlying LPs are the subject of further investigation. However, as shown in a separate article, for a large set of test problems in bimatrix games, not only the MMR payoffs for both players are better than any NE payoffs, so is the computing time of MMR in contrast with that of Lemke-Howsen algorithm. In large size problems the latter algorithm even fails to produce a Nash equilibrium. In summary, solving MMR provides a worthy approximation even if Nash equilibrium is shown to be computable in polynomial-time.

The Bimatrix Nash Equilibrium (NE) for $m \times n$ real matrices $R$ and $C$, denoted as the {\it Row} and {\it Column} players, is characterized as follows: Let $\Delta =S_m \times S_n$, where $S_k$ denotes the unit simplex in $\mathbb{R}^k$. For a given point $p=(x,y) \in \Delta$, define $R[p]=x^TRy$ and $C[p]=x^TCy$. Consequently, there exists a subset $\Delta_* \subset \Delta$ such that for any $p_*=(x_*,y_*) \in \Delta_*$, $\max_{p \in \Delta, y=y_*}R[p]=R[p_*]$ and $\max_{p \in \Delta, x=x_* } C[p]=C[p_*]$. The computational complexity of bimatrix NE falls within the class of {\it PPAD-complete}. Although the von Neumann Minimax Theorem is a special case of bimatrix NE, we introduce a novel extension termed {\it Trilinear Minimax Relaxation} (TMR) with the following implications: Let $\lambda^*=\min_{\alpha \in S_{2}} \max_{p \in \Delta} (\alpha_1 R[p]+ \alpha_2C[p])$ and $\lambda_*=\max_{p \in \Delta} \min_{\alpha \in S_{2}} (\alpha_1 R[p]+ \alpha_2C[p])$. $\lambda^* \geq \lambda_*$. $\lambda^*$ is computable as a linear programming in $O(mn)$ time, ensuring $\max_{p_* \in \Delta_*}\min \{R[p_*], C[p_*]\} \leq \lambda^*$, meaning that in any Nash Equilibrium it is not possible to have both players' payoffs to exceed $\lambda^*$. $\lambda^*=\lambda_*$ if and only if there exists $p^* \in \Delta$ such that $\lambda^*= \min\{R[p^*], C[p^*]\}$. Such a $p^*$ serves as an approximate Nash Equilibrium. We analyze the cases where such $p^*$ exists and is computable. Even when $\lambda^* > \lambda_*$, we derive approximate Nash Equilibria. In summary, the aforementioned properties of TMR and its efficient computational aspects underscore its significance and relevance for Nash Equilibrium, irrespective of the computational complexity associated with bimatrix Nash Equilibrium. Finally, we extend TMR to scenarios involving three or more players.

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $\Sigma$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.

Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of powers of the Coppersmith-Winograd tensor. In this paper we show how to use this new approach to improve the exponent of rectangular matrix multiplication as well. Our main technical contribution is showing how to combine this analysis of the combination loss and the analysis of the fourth power of the Coppersmith-Winograd tensor in the context of rectangular matrix multiplication developed by Le Gall and Urrutia (SODA 2018).

Secure multiparty computation (MPC) schemes allow two or more parties to conjointly compute a function on their private input sets while revealing nothing but the output. Existing state-of-the-art number-theoretic-based designs face the threat of attacks through quantum algorithms. In this context, we present secure MPC protocols that can withstand quantum attacks. We first present the design and analysis of an information-theoretic secure oblivious linear evaluation (OLE), namely ${\sf qOLE}$ in the quantum domain, and show that our ${\sf qOLE}$ is safe from external attacks. In addition, our scheme satisfies all the security requirements of a secure OLE. We further utilize ${\sf qOLE}$ as a building block to construct a quantum-safe multiparty private set intersection (MPSI) protocol.

2D-based Industrial Anomaly Detection has been widely discussed, however, multimodal industrial anomaly detection based on 3D point clouds and RGB images still has many untouched fields. Existing multimodal industrial anomaly detection methods directly concatenate the multimodal features, which leads to a strong disturbance between features and harms the detection performance. In this paper, we propose Multi-3D-Memory (M3DM), a novel multimodal anomaly detection method with hybrid fusion scheme: firstly, we design an unsupervised feature fusion with patch-wise contrastive learning to encourage the interaction of different modal features; secondly, we use a decision layer fusion with multiple memory banks to avoid loss of information and additional novelty classifiers to make the final decision. We further propose a point feature alignment operation to better align the point cloud and RGB features. Extensive experiments show that our multimodal industrial anomaly detection model outperforms the state-of-the-art (SOTA) methods on both detection and segmentation precision on MVTec-3D AD dataset. Code is available at //github.com/nomewang/M3DM.

Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.

We investigate a lattice-structured LSTM model for Chinese NER, which encodes a sequence of input characters as well as all potential words that match a lexicon. Compared with character-based methods, our model explicitly leverages word and word sequence information. Compared with word-based methods, lattice LSTM does not suffer from segmentation errors. Gated recurrent cells allow our model to choose the most relevant characters and words from a sentence for better NER results. Experiments on various datasets show that lattice LSTM outperforms both word-based and character-based LSTM baselines, achieving the best results.