Sparse Signal Recovery Using Markov Random Fields
Volkan Cevher, Marco Duarte, Chinmay Hedge, Richard Baraniuk
Compressive Sensing (CS) combines sampling and compression into a single sub-Nyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov Random Fields (MRFs) to represent sparse signals whose nonzero coefficients are clustered. Our new model-based reconstruction algorithm, dubbed Lattice Matching Pursuit (LaMP), stably recovers MRF-modeled signals using many fewer measurements and computations than the current state-of-the-art algorithms.
Measures of Clustering Quality: A Working Set of Axioms for Clustering Shai Ben-David, Margareta Ackerman
Aiming towards the development of a general clustering theory, we discuss abstract axiomatization for clustering. In this respect, we follow up on the work of Kleinberg, () that showed an impossibility result for such axiomatization. We argue that an impossibility result is not an inherent feature of clustering, but rather, to a large extent, it is an artifact of the specific formalism used in . As opposed to previous work focusing on clustering functions, we propose to address clustering quality measures as the object to be axiomatized. We show that principles like those formulated in Kleinberg’s axioms can be readily expressed in the latter framework without leading to inconsistency. A clustering-quality measure (CQM) is a function that, given a data set and its partition into clusters, returns a non-negative real number representing how strong or conclusive the clustering is. We analyze what clustering-quality measures should look like and introduce a set of requirements (axioms) for such measures. Our axioms capture the principles expressed by Kleinberg’s axioms while retaining consistency. We propose several natural clustering quality measures, all satisfying the proposed axioms. In addition, we analyze the computational complexity of evaluating the quality of a given clustering and show that, for the proposed CQMs, it can be computed in polynomial time.
Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning Ali Rahimi, Ben Recht
Randomized neural networks are immortalized in this well-known AI Koan: In the days when Sussman was a novice, Minsky once came to him as he sat hacking at the PDP-6. “What are you doing?” asked Minsky. “I am training a randomly wired neural net to play tic-tac-toe,” Sussman replied. “Why is the net wired randomly?” asked Minsky. Sussman replied, “I do not want it to have any preconceptions of how to play.”Minsky then shut his eyes. “Why do you close your eyes?” Sussman asked his teacher. “So that the room will be empty,” replied Minsky. At that moment, Sussman was enlightened. We analyze shallow random networks with the help of concentration of measure inequalities. Specifically, we consider architectures that compute a weighted sum of their inputs after passing them through a bank of arbitrary randomized nonlinearities. We identify conditions under which these networks exhibit good classification performance, and bound their test error in terms of the size of the dataset and the number of random nonlinearities.
Shape-Based Object Localization for Descriptive Classification Geremy Heitz, Gal Elidan, Benjamin Packer, Daphne Koller
Discriminative tasks, including object categorization and detection, are central components of high-level computer vision. Sometimes, however, we are inter- ested in more refined aspects of the object in an image, such as pose or particular regions. In this paper we develop a method (LOOPS) for learning a shape and image feature model that can be trained on a particular object class, and used to outline instances of the class in novel images. Furthermore, while the training data consists of uncorresponded outlines, the resulting LOOPS model contains a set of landmark points that appear consistently across instances, and can be accurately localized in an image. Our model achieves state-of-the-art results in precisely out- lining objects that exhibit large deformations and articulations in cluttered natural images. These localizations can then be used to address a range of tasks, including descriptive classification, search, and clustering.