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We introduce a representation of a 2D steady vector field ${{\mathbf v}}$ by two scalar fields $a$, $b$, such that the isolines of $a$ correspond to stream lines of ${{\mathbf v}}$, and $b$ increases with constant speed under integration of ${{\mathbf v}}$. This way, we get a direct encoding of stream lines, i.e., a numerical integration of ${{\mathbf v}}$ can be replaced by a local isoline extraction of $a$. To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields $a$ and $b$ can be computed. We show several evaluations on non-trivial vector fields.

Open set recognition (OSR) requires the model to classify samples that belong to closed sets while rejecting unknown samples during test. Currently, generative models often perform better than discriminative models in OSR, but recent studies show that generative models may be computationally infeasible or unstable on complex tasks. In this paper, we provide insights into OSR and find that learning supplementary representations can theoretically reduce the open space risk. Based on the analysis, we propose a new model, namely Multi-Expert Diverse Attention Fusion (MEDAF), that learns diverse representations in a discriminative way. MEDAF consists of multiple experts that are learned with an attention diversity regularization term to ensure the attention maps are mutually different. The logits learned by each expert are adaptively fused and used to identify the unknowns through the score function. We show that the differences in attention maps can lead to diverse representations so that the fused representations can well handle the open space. Extensive experiments are conducted on standard and OSR large-scale benchmarks. Results show that the proposed discriminative method can outperform existing generative models by up to 9.5% on AUROC and achieve new state-of-the-art performance with little computational cost. Our method can also seamlessly integrate existing classification models. Code is available at //github.com/Vanixxz/MEDAF.

In private computation, a user wishes to retrieve a function evaluation of messages stored on a set of databases without revealing the function's identity to the databases. Obead \emph{et al.} introduced a capacity outer bound for private nonlinear computation, dependent on the order of the candidate functions. Focusing on private \emph{quadratic monomial} computation, we propose three methods for ordering candidate functions: a graph edge-coloring method, a graph-distance method, and an entropy-based greedy method. We confirm, via an exhaustive search, that all three methods yield an optimal ordering for $f < 6$ messages. For $6 \leq f \leq 12$ messages, we numerically evaluate the performance of the proposed methods compared with a directed random search. For almost all scenarios considered, the entropy-based greedy method gives the smallest gap to the best-found ordering.

We are given a set $\mathcal{Z}=\{(R_1,s_1),\ldots, (R_n,s_n)\}$, where each $R_i$ is a \emph{range} in $\Re^d$, such as rectangle or ball, and $s_i \in [0,1]$ denotes its \emph{selectivity}. The goal is to compute a small-size \emph{discrete data distribution} $\mathcal{D}=\{(q_1,w_1),\ldots, (q_m,w_m)\}$, where $q_j\in \Re^d$ and $w_j\in [0,1]$ for each $1\leq j\leq m$, and $\sum_{1\leq j\leq m}w_j= 1$, such that $\mathcal{D}$ is the most \emph{consistent} with $\mathcal{Z}$, i.e., $\mathrm{err}_p(\mathcal{D},\mathcal{Z})=\frac{1}{n}\sum_{i=1}^n\! \lvert{s_i-\sum_{j=1}^m w_j\cdot 1(q_j\in R_i)}\rvert^p$ is minimized. In a database setting, $\mathcal{Z}$ corresponds to a workload of range queries over some table, together with their observed selectivities (i.e., fraction of tuples returned), and $\mathcal{D}$ can be used as compact model for approximating the data distribution within the table without accessing the underlying contents. In this paper, we obtain both upper and lower bounds for this problem. In particular, we show that the problem of finding the best data distribution from selectivity queries is $\mathsf{NP}$-complete. On the positive side, we describe a Monte Carlo algorithm that constructs, in time $O((n+\delta^{-d})\delta^{-2}\mathop{\mathrm{polylog}})$, a discrete distribution $\tilde{\mathcal{D}}$ of size $O(\delta^{-2})$, such that $\mathrm{err}_p(\tilde{\mathcal{D}},\mathcal{Z})\leq \min_{\mathcal{D}}\mathrm{err}_p(\mathcal{D},\mathcal{Z})+\delta$ (for $p=1,2,\infty$) where the minimum is taken over all discrete distributions. We also establish conditional lower bounds, which strongly indicate the infeasibility of relative approximations as well as removal of the exponential dependency on the dimension for additive approximations. This suggests that significant improvements to our algorithm are unlikely.

We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability, and up to an additive constant with polynomial probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. These results imply polynomial time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.

For a permutation $\pi:[k] \to [k]$, a function $f:[n] \to \mathbb{R}$ contains a $\pi$-appearance if there exists $1 \leq i_1 < i_2 < \dots < i_k \leq n$ such that for all $s,t \in [k]$, $f(i_s) < f(i_t)$ if and only if $\pi(s) < \pi(t)$. The function is $\pi$-free if it has no $\pi$-appearances. In this paper, we investigate the problem of testing whether an input function $f$ is $\pi$-free or whether $f$ differs on at least $\varepsilon n$ values from every $\pi$-free function. This is a generalization of the well-studied monotonicity testing and was first studied by Newman, Rabinovich, Rajendraprasad and Sohler (Random Structures and Algorithms 2019). We show that for all constants $k \in \mathbb{N}$, $\varepsilon \in (0,1)$, and permutation $\pi:[k] \to [k]$, there is a one-sided error $\varepsilon$-testing algorithm for $\pi$-freeness of functions $f:[n] \to \mathbb{R}$ that makes $\tilde{O}(n^{o(1)})$ queries. We improve significantly upon the previous best upper bound $O(n^{1 - 1/(k-1)})$ by Ben-Eliezer and Canonne (SODA 2018). Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.

We consider the performance of a least-squares regression model, as judged by out-of-sample $R^2$. Shapley values give a fair attribution of the performance of a model to its input features, taking into account interdependencies between features. Evaluating the Shapley values exactly requires solving a number of regression problems that is exponential in the number of features, so a Monte Carlo-type approximation is typically used. We focus on the special case of least-squares regression models, where several tricks can be used to compute and evaluate regression models efficiently. These tricks give a substantial speed up, allowing many more Monte Carlo samples to be evaluated, achieving better accuracy. We refer to our method as least-squares Shapley performance attribution (LS-SPA), and describe our open-source implementation.

Neural machine translation (NMT) is a deep learning based approach for machine translation, which yields the state-of-the-art translation performance in scenarios where large-scale parallel corpora are available. Although the high-quality and domain-specific translation is crucial in the real world, domain-specific corpora are usually scarce or nonexistent, and thus vanilla NMT performs poorly in such scenarios. Domain adaptation that leverages both out-of-domain parallel corpora as well as monolingual corpora for in-domain translation, is very important for domain-specific translation. In this paper, we give a comprehensive survey of the state-of-the-art domain adaptation techniques for NMT.

We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.