Principal Component Analysis (PCA) is one of the most important unsupervised methods to handle high-dimensional data. However, due to the high computational complexity of its eigen decomposition solution, it hard to apply PCA to the large-scale data with high dimensionality. Meanwhile, the squared L2-norm based objective makes it sensitive to data outliers. In recent research, the L1-norm maximization based PCA method was proposed for efficient computation and being robust to outliers. However, this work used a greedy strategy to solve the eigen vectors. Moreover, the L1-norm maximization based objective may not be the correct robust PCA formulation, because it loses the theoretical connection to the minimization of data reconstruction error, which is one of the most important intuitions and goals of PCA. In this paper, we propose to maximize the L21-norm based robust PCA objective, which is theoretically connected to the minimization of reconstruction error. More importantly, we propose the efficient non-greedy optimization algorithms to solve our objective and the more general L21-norm maximization problem with theoretically guaranteed convergence. Experimental results on real world data sets show the effectiveness of the proposed method for principal component analysis.
Capital University of Economics and Business（首都经济贸易大学）；Hainan University（海南大学）；China Agricultural University（中国农业大学）
Classical principal component analysis (PCA) may suffer from the sensitivity to outliers and noise. Therefore PCA based on $\ell_1$-norm and $\ell_p$-norm ($0 < p < 1$) have been studied. Among them, the ones based on $\ell_p$-norm seem to be most interesting from the robustness point of view. However, their numerical performance is not satisfactory. Note that, although T$\ell_1$-norm is similar to $\ell_p$-norm ($0 < p < 1$) in some sense, it has the stronger suppression effect to outliers and better continuity. So PCA based on T$\ell_1$-norm is proposed in this paper. Our numerical experimen…
Singapore University of Technology and Design（新加坡科技设计大学）；University of Edinburgh（爱丁堡大学）
We propose a novel topic discovery algorithm for unlabeled images based on the bag-of-words (BoW) framework. We first extract a dictionary of visual words and subsequently for each image compute a visual word occurrence histogram. We view these histograms as rows of a large matrix from which we extract sparse principal components (PCs). Each PC identifies a sparse combination of visual words which co-occur frequently in some images but seldom appear in others. Each sparse PC corresponds to a topic, and images whose interference with the PC is high belong to that topic, revealing the common par…
Graduate School of Information Sciences and Technology（信息科学与技术研究生院）；The University of Tokyo（东京大学）
Motivated by, e.g., sensitivity analysis and end-to-end learning, the demand for differentiable optimization algorithms has been significantly increasing. In this paper, we establish a theoretically guaranteed versatile framework that makes the greedy algorithm for monotone submodular function maximization differentiable. We smooth the greedy algorithm via randomization, and prove that it almost recovers original approximation guarantees in expectation for the cases of cardinality and $κ$-extensible system constrains. We also show how to efficiently compute unbiased gradient estimators of any …
Boston University（波士顿大学）；Northeastern University（东北大学）
We study parallel algorithms for the problem of maximizing a non-negative submodular function. Our main result is an algorithm that achieves a nearly-optimal $1/2 -ε$ approximation using $O(\log(1/ε) / ε)$ parallel rounds of function evaluations. Our algorithm is based on a continuous variant of the double greedy algorithm of Buchbinder et al. that achieves the optimal $1/2$ approximation in the sequential setting. Our algorithm applies more generally to the problem of maximizing a continuous diminishing-returns (DR) function.
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Carnegie Mellon University（卡内基梅隆大学）
Multimodal learning considers learning from multi-modality data, aiming to fuse heterogeneous sources of information. However, it is not always feasible to leverage all available modalities due to memory constraints. Further, training on all the modalities may be inefficient when redundant information exists within data, such as different subsets of modalities providing similar performance. In light of these challenges, we study modality selection, intending to efficiently select the most informative and complementary modalities under certain computational constraints. We formulate a theoretic…
University of Amsterdam（阿姆斯特丹大学）；University of Oxford（牛津大学）；University of Liverpool（利物浦大学）
Submodular function maximization finds application in a variety of real-world decision-making problems. However, most existing methods, based on greedy maximization, assume it is computationally feasible to evaluate F, the function being maximized. Unfortunately, in many realistic settings F is too expensive to evaluate exactly even once. We present probably approximately correct greedy maximization, which requires access only to cheap anytime confidence bounds on F and uses them to prune elements. We show that, with high probability, our method returns an approximately optimal set. We propose…
Open University of Israel（以色列开放大学）；Yale University（耶鲁大学）
It is known that greedy methods perform well for maximizing monotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show - arguably, surprisingly - that invoking the classical greedy algorithm $O(\sqrt{k})$-times leads to the (currently) fastest deterministic algorithm, called Repeated Greedy, for maximizing a general submodular function subject to $k$-independent system constraints. Repeated Greedy achieves $(1 + O(1/\sqrt{k}))k$ approximation using $O(nr\sqrt{k})$ function evaluations (here, $n$ and $r$ denote the size o…
KAUST（阿卜杜拉国王科技大学）；Purdue University（普渡大学）；Iowa State University（爱荷华州立大学）
We investigate the problem of unconstrained combinatorial multi-armed bandits with full-bandit feedback and stochastic rewards for submodular maximization. Previous works investigate the same problem assuming a submodular and monotone reward function. In this work, we study a more general problem, i.e., when the reward function is not necessarily monotone, and the submodularity is assumed only in expectation. We propose Randomized Greedy Learning (RGL) algorithm and theoretically prove that it achieves a $\frac{1}{2}$-regret upper bound of $\tilde{\mathcal{O}}(n T^{\frac{2}{3}})$ for horizon $…
We propose and analyze batch greedy heuristics for cardinality constrained maximization of non-submodular non-decreasing set functions. We consider the standard greedy paradigm, along with its distributed greedy and stochastic greedy variants. Our theoretical guarantees are characterized by the combination of submodularity and supermodularity ratios. We argue how these parameters define tight modular bounds based on incremental gains, and provide a novel reinterpretation of the classical greedy algorithm using the minorize-maximize (MM) principle. Based on that analogy, we propose a new class …
We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of …
Purdue University（普渡大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Many problems in signal processing and machine learning can be formalized as weak submodular optimization tasks. For such problems, a simple greedy algorithm (\textsc{Greedy}) is guaranteed to find a solution achieving the objective with a value no worse than $1-e^{-1/c}$ of the optimal, where $c$ is the multiplicative weak-submodularity constant. Due to the high cost of querying large-scale systems, the complexity of \textsc{Greedy} becomes prohibitive in contemporary applications. In this work, we study the tradeoff between performance and complexity when one resorts to random sampling strat…
Yale University（耶鲁大学）；The Open University of Israel（以色列开放大学）
Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for set functions generalizing submodular functions, has been the focus of recent works. One such class, known as weakly submodular functions, has received a lot of attention. A key result proved by Das and Kempe (2011) showed that the approximation ratio…
Department of Computer Science, Ben-Gurion University of the Negev（本·古里安大学计算机科学系）；Department of Computing and Software, McMaster University（麦斯特大学计算与软件系）；Canada CIFAR AI Chair, Vector Institute（加拿大 CIFAR 人工智能主席，向量研究所）
Hasso Plattner Institute（哈索普拉特纳研究所）；The University of Adelaide（阿德莱德大学）
We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature. In contrast to other heuristics, this requires no problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size. In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings. Additionally, we study this problem …
Sharif University（谢里夫大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodu…
We propose a streaming submodular maximization algorithm "stream clipper" that performs as well as the offline greedy algorithm on document/video summarization in practice. It adds elements from a stream either to a solution set $S$ or to an extra buffer $B$ based on two adaptive thresholds, and improves $S$ by a final greedy step that starts from $S$ adding elements from $B$. During this process, swapping elements out of $S$ can occur if doing so yields improvements. The thresholds adapt based on if current memory utilization exceeds a budget, e.g., it increases the lower threshold, and remov…
Profit maximization (PM) is to select a subset of users as seeds for viral marketing in online social networks, which balances between the cost and the profit from influence spread. We extend PM to that under the general marketing strategy, and form continuous profit maximization (CPM-MS) problem, whose domain is on integer lattices. The objective function of our CPM-MS is dr-submodular, but non-monotone. It is a typical case of unconstrained dr-submodular maximization (UDSM) problem, and take it as a starting point, we study UDSM systematically in this paper, which is very different from thos…
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）
In this paper, we study the non-monotone adaptive submodular maximization problem subject to a cardinality constraint. We first revisit the adaptive random greedy algorithm proposed in \citep{gotovos2015non}, where they show that this algorithm achieves a $1/e$ approximation ratio if the objective function is adaptive submodular and pointwise submodular. It is not clear whether the same guarantee holds under adaptive submodularity (without resorting to pointwise submodularity) or not. Our first contribution is to show that the adaptive random greedy algorithm achieves a $1/e$ approximation rat…
Indiana University, Bloomington（印第安纳大学伯明顿分校）；The Alan Turing Institute（艾伦·图灵研究所）；Carnegie Mellon University（卡内基梅隆大学）
The problem of selecting a small-size representative summary of a large dataset is a cornerstone of machine learning, optimization and data science. Motivated by applications to recommendation systems and other scenarios with query-limited access to vast amounts of data, we propose a new rigorous algorithmic framework for a standard formulation of this problem as a submodular maximization subject to a linear (knapsack) constraint. Our framework is based on augmenting all partial Greedy solutions with the best additional item. It can be instantiated with negligible overhead in any model of comp…
MIT（麻省理工学院）；TTIC（丰田理工学院）；University of Washington（华盛顿大学）
``Composable core-sets'' are an efficient framework for solving optimization problems in massive data models. In this work, we consider efficient construction of composable core-sets for the determinant maximization problem. This can also be cast as the MAP inference task for determinantal point processes, that have recently gained a lot of interest for modeling diversity and fairness. The problem was recently studied in [IMOR'18], where they designed composable core-sets with the optimal approximation bound of $\tilde O(k)^k$. On the other hand, the more practical Greedy algorithm has been pr…
We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy (CDCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using on…
We study the problem of maximizing a monotone set function subject to a cardinality constraint $k$ in the setting where some number of elements $τ$ is deleted from the returned set. The focus of this work is on the worst-case adversarial setting. While there exist constant-factor guarantees when the function is submodular, there are no guarantees for non-submodular objectives. In this work, we present a new algorithm Oblivious-Greedy and prove the first constant-factor approximation guarantees for a wider class of non-submodular objectives. The obtained theoretical bounds are the first constan…
We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $τ$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $τ= o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $τ= o(k)$. Our algorithm constructs partitions consisting of buckets…
A variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity - number of instances that can fit in memory - must grow with the data set size. In practice, while one can provision many machines, the capacity of each machine is limited by physical constraints. We propose a truly scalable approach for distributed submodular maximization under fixed capacity. The proposed framework applies to a broad class of algorithms and constraints and provid…
Inference for latent feature models is inherently difficult as the inference space grows exponentially with the size of the input data and number of latent features. In this work, we use Kurihara & Welling (2008)'s maximization-expectation framework to perform approximate MAP inference for linear-Gaussian latent feature models with an Indian Buffet Process (IBP) prior. This formulation yields a submodular function of the features that corresponds to a lower bound on the model evidence. By adding a constant to this function, we obtain a nonnegative submodular function that can be maximized via …
University of Coimbra（科英布拉大学）；Institute of Systems and Robotics（系统与机器人研究所）
This paper presents a novel feature selection method based on the conditional mutual information (CMI). The proposed High Order Conditional Mutual Information Maximization (HOCMIM) incorporates high order dependencies into the feature selection procedure and has a straightforward interpretation due to its bottom-up derivation. The HOCMIM is derived from the CMI's chain expansion and expressed as a maximization optimization problem. The maximization problem is solved using a greedy search procedure, which speeds up the entire feature selection process. The experiments are run on a set of benchm…
Microsoft Research Cambridge（微软研究院剑桥）；ETH Zurich（苏黎世联邦理工学院）
We consider learning of submodular functions from data. These functions are important in machine learning and have a wide range of applications, e.g. data summarization, feature selection and active learning. Despite their combinatorial nature, submodular functions can be maximized approximately with strong theoretical guarantees in polynomial time. Typically, learning the submodular function and optimization of that function are treated separately, i.e. the function is first learned using a proxy objective and subsequently maximized. In contrast, we show how to perform learning and optimizati…
Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）；Université d’Angers（昂热大学）；iAdvize（iAdvize公司）
When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. Principal curves act as a nonlinear generalization of PCA and the present paper proposes a novel algorithm to automatically and sequentially learn principal curves from data streams. We show that our procedure is supported by regret bounds with optimal sublinear remainder terms. A greedy local search implementation (called \texttt{slpc}, for Sequential Learning Principal Curves) that incorporates both sleeping experts and multi-armed bandit ingre…
