Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks

We discuss the computational complexity of approximating maximum a posteriori inference in sum-product networks. We first show NP-hardness in trees of height two by a reduction from maximum independent set; this implies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to approximate the problem within a factor $2^{f(n)}$ for any sublinear function $f$ of the size of the input $n$. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor $2^{c \cdot n}$ for some constant $c < 1$. Last, we present a simple algorithm, and show that it provably produces solutions at least as good as, and potentially much better than, the max-product algorithm. We empirically analyze the proposed algorithm against max-product using synthetic and realistic networks.Comment: 18 page

Learning Tractable Probabilistic Models for Fault Localization

In recent years, several probabilistic techniques have been applied to various debugging problems. However, most existing probabilistic debugging systems use relatively simple statistical models, and fail to generalize across multiple programs. In this work, we propose Tractable Fault Localization Models (TFLMs) that can be learned from data, and probabilistically infer the location of the bug. While most previous statistical debugging methods generalize over many executions of a single program, TFLMs are trained on a corpus of previously seen buggy programs, and learn to identify recurring patterns of bugs. Widely-used fault localization techniques such as TARANTULA evaluate the suspiciousness of each line in isolation; in contrast, a TFLM defines a joint probability distribution over buggy indicator variables for each line. Joint distributions with rich dependency structure are often computationally intractable; TFLMs avoid this by exploiting recent developments in tractable probabilistic models (specifically, Relational SPNs). Further, TFLMs can incorporate additional sources of information, including coverage-based features such as TARANTULA. We evaluate the fault localization performance of TFLMs that include TARANTULA scores as features in the probabilistic model. Our study shows that the learned TFLMs isolate bugs more effectively than previous statistical methods or using TARANTULA directly.Comment: Fifth International Workshop on Statistical Relational AI (StaR-AI 2015

The Libra Toolkit for Probabilistic Models

The Libra Toolkit is a collection of algorithms for learning and inference with discrete probabilistic models, including Bayesian networks, Markov networks, dependency networks, and sum-product networks. Compared to other toolkits, Libra places a greater emphasis on learning the structure of tractable models in which exact inference is efficient. It also includes a variety of algorithms for learning graphical models in which inference is potentially intractable, and for performing exact and approximate inference. Libra is released under a 2-clause BSD license to encourage broad use in academia and industry

Depth Estimation via Affinity Learned with Convolutional Spatial Propagation Network

Depth estimation from a single image is a fundamental problem in computer vision. In this paper, we propose a simple yet effective convolutional spatial propagation network (CSPN) to learn the affinity matrix for depth prediction. Specifically, we adopt an efficient linear propagation model, where the propagation is performed with a manner of recurrent convolutional operation, and the affinity among neighboring pixels is learned through a deep convolutional neural network (CNN). We apply the designed CSPN to two depth estimation tasks given a single image: (1) To refine the depth output from state-of-the-art (SOTA) existing methods; and (2) to convert sparse depth samples to a dense depth map by embedding the depth samples within the propagation procedure. The second task is inspired by the availability of LIDARs that provides sparse but accurate depth measurements. We experimented the proposed CSPN over two popular benchmarks for depth estimation, i.e. NYU v2 and KITTI, where we show that our proposed approach improves in not only quality (e.g., 30% more reduction in depth error), but also speed (e.g., 2 to 5 times faster) than prior SOTA methods.Comment: 14 pages, 8 figures, ECCV 201