Data clustering is a fundamental problem with a wide range of applications. Standard methods, eg the $k$-means method, usually require solving a non-convex optimization problem. Recently, total variation based convex relaxation to the $k$-means model has emerged as an attractive alternative for data clustering. However, the existing results on its exact clustering property, ie, the condition imposed on data so that the method can provably give correct identification of all cluster memberships, is only applicable to very specific data and is also much more restrictive than that of some other methods. This paper aims at the revisit of total variation based convex clustering, by proposing a weighted sum-of-$\ell_1$-norm relating convex model. Its exact clustering property established in this paper, in both deterministic and probabilistic context, is applicable to general data and is much sharper than the existing results. These results provided good insights to advance the research on convex clustering. Moreover, the experiments also demonstrated that the proposed convex model has better empirical performance when be compared to standard clustering methods, and thus it can see its potential in practice.

When consequential decisions are informed by algorithmic input, individuals may feel compelled to alter their behavior in order to gain a system's approval. Previous models of agent responsiveness, termed "strategic manipulation," have analyzed the interaction between a learner and agents in a world where all agents are equally able to manipulate their features in an attempt to "trick" a published classifier. In cases of real world classification, however, an agent's ability to adapt to an algorithm, is not simply a function of her personal interest in receiving a positive classification, but is bound up in a complex web of social factors that affect her ability to pursue certain action responses. In this paper, we adapt models of strategic manipulation to better capture dynamics that may arise in a setting of social inequality wherein candidate groups face different costs to manipulation. We find that whenever one group's costs are higher than the other's, the learner's equilibrium strategy exhibits an inequality-reinforcing phenomenon wherein the learner erroneously admits some members of the advantaged group, while erroneously excluding some members of the disadvantaged group. We also consider the effects of potential interventions in which a learner can subsidize members of the disadvantaged group, lowering their costs in order to improve her own classification performance. Here we encounter a paradoxical result: there exist cases in which providing a subsidy improves only the learner's utility while actually making both candidate groups worse-off--even the group receiving the subsidy. Our results reveal the potentially adverse social ramifications of deploying tools that attempt to evaluate an individual's "quality" when agents' capacities to adaptively respond differ.

The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i) the mean-field approximation with roots in statistical physics, which estimates the free energy from below, and (ii) hierarchies of convex relaxations with roots in theoretical computer science, which estimate the free energy from above. We show, surprisingly, that the tight regime for both methods to compute the free energy to leading order is identical. More precisely, we show that the mean-field approximation is within $O((n\|J\|_{F})^{2/3})$ of the free energy, where $\|J\|_F$ denotes the Frobenius norm of the interaction matrix of the Ising model. This simultaneously subsumes both the breakthrough work of Basak and Mukherjee, who showed the tight result that the mean-field approximation is within $o(n)$ whenever $\|J\|_{F} = o(\sqrt{n})$, as well as the work of Jain, Koehler, and Mossel, who gave the previously best known non-asymptotic bound of $O((n\|J\|_{F})^{2/3}\log^{1/3}(n\|J\|_{F}))$. We give a simple, algorithmic proof of this result using a convex relaxation proposed by Risteski based on the Sherali-Adams hierarchy, automatically giving sub-exponential time approximation schemes for the free energy in this entire regime. Our algorithmic result is tight under Gap-ETH. We furthermore combine our techniques with spin glass theory to prove (in a strong sense) the optimality of correlation rounding, refuting a recent conjecture of Allen, O'Donnell, and Zhou. Finally, we give the tight generalization of all of these results to $k$-MRFs, capturing as a special case previous work on approximating MAX-$k$-CSP.

Recently, considerable effort has been devoted to deep domain adaptation in computer vision and machine learning communities. However, most of existing work only concentrates on learning shared feature representation by minimizing the distribution discrepancy across different domains. Due to the fact that all the domain alignment approaches can only reduce, but not remove the domain shift. Target domain samples distributed near the edge of the clusters, or far from their corresponding class centers are easily to be misclassified by the hyperplane learned from the source domain. To alleviate this issue, we propose to joint domain alignment and discriminative feature learning, which could benefit both domain alignment and final classification. Specifically, an instance-based discriminative feature learning method and a center-based discriminative feature learning method are proposed, both of which guarantee the domain invariant features with better intra-class compactness and inter-class separability. Extensive experiments show that learning the discriminative features in the shared feature space can significantly boost the performance of deep domain adaptation methods.

Matrix factorization is a simple and effective solution to the recommendation problem. It has been extensively employed in the industry and has attracted much attention from the academia. However, it is unclear what the low-dimensional matrices represent. We show that matrix factorization can actually be seen as simultaneously calculating the eigenvectors of the user-user and item-item sample co-occurrence matrices. We then use insights from random matrix theory (RMT) to show that picking the top eigenvectors corresponds to removing sampling noise from user/item co-occurrence matrices. Therefore, the low-dimension matrices represent a reduced noise user and item co-occurrence space. We also analyze the structure of the top eigenvector and show that it corresponds to global effects and removing it results in less popular items being recommended. This increases the diversity of the items recommended without affecting the accuracy.

Extractive summarization models require sentence-level labels, which are usually created heuristically (e.g., with rule-based methods) given that most summarization datasets only have document-summary pairs. Since these labels might be suboptimal, we propose a latent variable extractive model where sentences are viewed as latent variables and sentences with activated variables are used to infer gold summaries. During training the loss comes \emph{directly} from gold summaries. Experiments on the CNN/Dailymail dataset show that our model improves over a strong extractive baseline trained on heuristically approximated labels and also performs competitively to several recent models.

Autonomous aerial cinematography has the potential to enable automatic capture of aesthetically pleasing videos without requiring human intervention, empowering individuals with the capability of high-end film studios. Current approaches either only handle off-line trajectory generation, or offer strategies that reason over short time horizons and simplistic representations for obstacles, which result in jerky movement and low real-life applicability. In this work we develop a method for aerial filming that is able to trade off shot smoothness, occlusion, and cinematography guidelines in a principled manner, even under noisy actor predictions. We present a novel algorithm for real-time covariant gradient descent that we use to efficiently find the desired trajectories by optimizing a set of cost functions. Experimental results show that our approach creates attractive shots, avoiding obstacles and occlusion 65 times over 1.25 hours of flight time, re-planning at 5 Hz with a 10 s time horizon. We robustly film human actors, cars and bicycles performing different motion among obstacles, using various shot types.

This letter is concerned with a recently developed paradigm of population-based optimization, termed particle filter optimization (PFO). In contrast with the commonly used meta-heuristics based methods, the PFO paradigm is attractive in terms of coherence in theory and easiness in mathematical analysis and interpretation. However, current PFO algorithms only work for single-objective optimization cases, while many real-life problems involve multiple objectives to be optimized simultaneously. To this end, we make an effort to extend the scope of application of the PFO paradigm to multi-objective optimization (MOO) cases. An idea called path sampling is adopted within the PFO scheme to balance the different objectives to be optimized. The resulting algorithm is thus termed PFO with Path Sampling (PFOPS). Experimental results show that the proposed algorithm works consistently well for three different types of MOO problems, which are characterized by an associated convex, concave and discontinuous Pareto front, respectively.

A challenge in creating a dataset for machine reading comprehension (MRC) is to collect questions that require a sophisticated understanding of language to answer beyond using superficial cues. In this work, we investigate what makes questions easier across recent 12 MRC datasets with three question styles (answer extraction, description, and multiple choice). We propose to employ simple heuristics to split each dataset into easy and hard subsets and examine the performance of two baseline models for each of the subsets. We then manually annotate questions sampled from each subset with both validity and requisite reasoning skills to investigate which skills explain the difference between easy and hard questions. From this study, we observed that (i) the baseline performances for the hard subsets remarkably degrade compared to those of entire datasets, (ii) hard questions require knowledge inference and multiple-sentence reasoning in comparison with easy questions, and (iii) multiple-choice questions tend to require a broader range of reasoning skills than answer extraction and description questions. These results suggest that one might overestimate recent advances in MRC.

One way to interpret neural model predictions is to highlight the most important input features---for example, a heatmap visualization over the words in an input sentence. In existing interpretation methods for NLP, a word's importance is determined by either input perturbation---measuring the decrease in model confidence when that word is removed---or by the gradient with respect to that word. To understand the limitations of these methods, we use input reduction, which iteratively removes the least important word from the input. This exposes pathological behaviors of neural models: the remaining words appear nonsensical to humans and are not the ones determined as important by interpretation methods. As we confirm with human experiments, the reduced examples lack information to support the prediction of any label, but models still make the same predictions with high confidence. To explain these counterintuitive results, we draw connections to adversarial examples and confidence calibration: pathological behaviors reveal difficulties in interpreting neural models trained with maximum likelihood. To mitigate their deficiencies, we fine-tune the models by encouraging high entropy outputs on reduced examples. Fine-tuned models become more interpretable under input reduction without accuracy loss on regular examples.