We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. This paper extends and clarifies the scope of scenarios for which blind source separation becomes meaningful.
Also conceptually related is blind deconvolution (see e.g., @cite_33 @cite_19 ), however, the convolution model makes analysis and identifiability severely different @cite_26 .
@cite_19 @cite_26 @cite_33
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Identifiability is a key concern in ill-posed blind deconvolution problems arising in wireless communications and image processing. The single channel version of the problem is the most challenging and there have been efforts to use sparse models for regularizing the problem. Identifiability of the sparse blind deconvolution problem is analyzed and it is established that a simple sparsity assumption in the canonical basis is insufficient for unique recovery; a surprising negative result. The proof technique involves lifting the deconvolution problem into a rank one matrix recovery problem and analyzing the rank two nullspace of the resultant linear operator. A DoF (degrees of freedom) wise tight parametrized subset of this rank two null-space is constructed to establish the results. Blind deconvolution (BD), the resolution of a signal and a filter given their convolution, arises in many applications. Without further constraints, BD is ill-posed. In practice, subspace or sparsity constraints have been imposed to reduce the search space, and have shown some empirical success. However, existing theoretical analysis on uniqueness in BD is rather limited. As an effort to address the still mysterious question, we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution, subject to subspace or sparsity constraints. These sufficient conditions provide the first algebraic sample complexities for BD. We first derive a sufficient condition that applies to almost all bases or frames. For blind deconvolution of vectors in @math , with two subspace constraints of dimensions @math and @math , the required sample complexity is @math . Then we impose a sub-band structure on one basis, and derive a sufficient condition that involves a relaxed sample complexity @math , which we show to be optimal. We present the extensions of these results to BD with sparsity constraints or mixed constraints, with the sparsity level replacing the subspace dimension. The cost for the unknown support in this case is an extra factor of 2 in the sample complexity. Recent approaches for instance-aware semantic labeling have augmented convolutional neural networks (CNNs) with complex multi-task architectures or computationally expensive graphical models. We present a method that leverages a fully convolutional network (FCN) to predict semantic labels, depth and an instance-based encoding using each pixel’s direction towards its corresponding instance center. Subsequently, we apply low-level computer vision techniques to generate state-of-the-art instance segmentation on the street scene datasets KITTI and Cityscapes. Our approach outperforms existing works by a large margin and can additionally predict absolute distances of individual instances from a monocular image as well as a pixel-level semantic labeling. We propose a general framework called Network Dissection for quantifying the interpretability of latent representations of CNNs by evaluating the alignment between individual hidden units and a set of semantic concepts. Given any CNN model, the proposed method draws on a data set of concepts to score the semantics of hidden units at each intermediate convolutional layer. The units with semantics are labeled across a broad range of visual concepts including objects, parts, scenes, textures, materials, and colors. We use the proposed method to test the hypothesis that interpretability is an axis-independent property of the representation space, then we apply the method to compare the latent representations of various networks when trained to solve different classification problems. We further analyze the effect of training iterations, compare networks trained with different initializations, and measure the effect of dropout and batch normalization on the interpretability of deep visual representations. We demonstrate that the proposed method can shed light on characteristics of CNN models and training methods that go beyond measurements of their discriminative power.