Let @math be a polynomial with at most @math non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on @math that runs in time polynomial in @math and @math and succeeds if the function satisfies the unique sign property: there is one output value which corresponds to a unique set of values of the participating parities. This sufficient condition is satisfied when every coefficient of f is perturbed by a small random noise, or satisfied with high probability when s parity functions are chosen randomly or when all the coefficients are positive. Learning sparse polynomials over the Boolean domain in time polynomial in @math and @math is considered notoriously hard in the worst-case. Our result shows that the problem is tractable for almost all sparse polynomials. Then, we show an application of this result to hypergraph sketching which is the problem of learning a sparse (both in the number of hyperedges and the size of the hyperedges) hypergraph from uniformly drawn random cuts. We also provide experimental results on a real world dataset.
On the practical side, we give an application of the theory to the problem of hypergraph sketching. We generalize a prior work @cite_6 that applied the compressed sensing approach discussed before to graph sketching on evolving social network graphs. In our algorithm, while the sample complexity requirements are higher, the time complexity is greatly reduced in comparison. We test our algorithms on a real dataset and show that the algorithm is able to scale well on sparse hypergraphs created out of Yahoo! messenger dataset by filtering through time and location stamps.
@cite_6
2973707709 1592346261 2123557958 1978676170
In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n2)-dimensional, we show the existence of a distribution over random projections into d-dimensional "sketch" space (d<< n2) such that the relevant properties of the original graph can be inferred from the sketch with high probability. Specifically, we show that: 1. d = O(n · polylog n) suffices to evaluate properties including connectivity, k-connectivity, bipartiteness, and to return any constant approximation of the weight of the minimum spanning tree. 2. d = O(n1+γ) suffices to compute graph sparsifiers, the exact MST, and approximate the maximum weighted matchings if we permit O(1 γ)-round adaptive sketches, i.e., a sequence of projections where each projection may be chosen dependent on the outcome of earlier sketches. Our results have two main applications, both of which have the potential to give rise to fruitful lines of further research. First, our results can be thought of as giving the first compressed-sensing style algorithms for graph data. Secondly, our work initiates the study of dynamic graph streams. There is already extensive literature on processing massive graphs in the data-stream model. However, the existing work focuses on graphs defined by a sequence of inserted edges and does not consider edge deletions. We think this is a curious omission given the existing work on both dynamic graphs in the non-streaming setting and dynamic geometric streaming. Our results include the first dynamic graph semi-streaming algorithms for connectivity, spanning trees, sparsification, and matching problems. In this paper, we study the problem of how to segment a freehand sketch at the object level. By carefully considering the basic principles of human perceptual organization, a real-time solution is presented to automatically segment a user's sketch during his her drawing. First, a graph-based sketch segmentation algorithm is proposed to segment a cluttered sketch into multiple parts based on the factor of proximity. Then, to improve the ability of detecting semantically meaningful objects, a semantic-based approach is introduced to simulate the past experience in the perceptual system by leveraging a web-scale clipart database. Finally, other important factors learnt from past experience, such as similarity, symmetry, direction, and closure, are also taken into account to make the approach more robust and practical. The proposed sketch segmentation framework has ability to handle complex sketches with overlapped objects. Extensive experimental results show the effectiveness of the proposed framework and algorithms. Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1--e)-approximation for Max-Cut must use n 1-O(e) space; moreover, beating 4 5-approximation requires polynomial space. For the sketching model, we show that every r-uniform hypergraph admits a (1+ e)-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(e-2n(r+log n)) edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems).