Performance bounds are given for exploratory co-clustering blockmodeling of bipartite graph data, where we assume the rows and columns of the data matrix are samples from an arbitrary population. This is equivalent to assuming that the data is generated from a nonsmooth graphon. It is shown that co-clusters found by any method can be extended to the row and column populations, or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon, with estimation error bounded by @math . Analogous performance bounds are also given for degree-corrected blockmodels and random dot product graphs, with error rates depending on the dimensionality of the latent variable space.
In @cite_30 and in this present paper, which consider only bipartite graphs, the emphasis is on exploratory analysis. Hence no assumptions are placed on the generative graphon. Unlike the works which assume smoothness or low rank structure, the object of inference is not the generative model itself, but rather a blocked version of it (this is defined precisely in Section ). This is reminiscent of some results for confidence intervals in nonparametric regression, in which the interval is centered not on the generative function or density itself, but rather on a smoothed or histogram-ed version [Sec 5.7 and Thm 6.20] wasserman2006all . The present paper can be viewed as a substantial improvement over @cite_30 ; for example, Theorem improves the rates of convergence from @math to @math , and also applies to computationally efficient estimates.
@cite_30
2056860348 2951523806 648143168 2964268978
In this paper we present a Bayesian framework for parsing images into their constituent visual patterns. The parsing algorithm optimizes the posterior probability and outputs a scene representation as a "parsing graph", in a spirit similar to parsing sentences in speech and natural language. The algorithm constructs the parsing graph and re-configures it dynamically using a set of moves, which are mostly reversible Markov chain jumps. This computational framework integrates two popular inference approaches--generative (top-down) methods and discriminative (bottom-up) methods. The former formulates the posterior probability in terms of generative models for images defined by likelihood functions and priors. The latter computes discriminative probabilities based on a sequence (cascade) of bottom-up tests filters. In our Markov chain algorithm design, the posterior probability, defined by the generative models, is the invariant (target) probability for the Markov chain, and the discriminative probabilities are used to construct proposal probabilities to drive the Markov chain. Intuitively, the bottom-up discriminative probabilities activate top-down generative models. In this paper, we focus on two types of visual patterns--generic visual patterns, such as texture and shading, and object patterns including human faces and text. These types of patterns compete and cooperate to explain the image and so image parsing unifies image segmentation, object detection, and recognition (if we use generic visual patterns only then image parsing will correspond to image segmentation (Tu and Zhu, 2002. IEEE Trans. PAMI, 24(5):657--673). We illustrate our algorithm on natural images of complex city scenes and show examples where image segmentation can be improved by allowing object specific knowledge to disambiguate low-level segmentation cues, and conversely where object detection can be improved by using generic visual patterns to explain away shadows and occlusions. In this paper we introduce a generative parametric model capable of producing high quality samples of natural images. Our approach uses a cascade of convolutional networks within a Laplacian pyramid framework to generate images in a coarse-to-fine fashion. At each level of the pyramid, a separate generative convnet model is trained using the Generative Adversarial Nets (GAN) approach (). Samples drawn from our model are of significantly higher quality than alternate approaches. In a quantitative assessment by human evaluators, our CIFAR10 samples were mistaken for real images around 40 of the time, compared to 10 for samples drawn from a GAN baseline model. We also show samples from models trained on the higher resolution images of the LSUN scene dataset. In this paper we introduce a generative parametric model capable of producing high quality samples of natural images. Our approach uses a cascade of convolutional networks within a Laplacian pyramid framework to generate images in a coarse-to-fine fashion. At each level of the pyramid, a separate generative convnet model is trained using the Generative Adversarial Nets (GAN) approach [11]. Samples drawn from our model are of significantly higher quality than alternate approaches. In a quantitative assessment by human evaluators, our CIFAR10 samples were mistaken for real images around 40 of the time, compared to 10 for samples drawn from a GAN baseline model. We also show samples from models trained on the higher resolution images of the LSUN scene dataset. As a new way of training generative models, Generative Adversarial Net (GAN) that uses a discriminative model to guide the training of the generative model has enjoyed considerable success in generating real-valued data. However, it has limitations when the goal is for generating sequences of discrete tokens. A major reason lies in that the discrete outputs from the generative model make it difficult to pass the gradient update from the discriminative model to the generative model. Also, the discriminative model can only assess a complete sequence, while for a partially generated sequence, it is nontrivial to balance its current score and the future one once the entire sequence has been generated. In this paper, we propose a sequence generation framework, called SeqGAN, to solve the problems. Modeling the data generator as a stochastic policy in reinforcement learning (RL), SeqGAN bypasses the generator differentiation problem by directly performing gradient policy update. The RL reward signal comes from the GAN discriminator judged on a complete sequence, and is passed back to the intermediate state-action steps using Monte Carlo search. Extensive experiments on synthetic data and real-world tasks demonstrate significant improvements over strong baselines.