OpenRaiser/Intern-Atlas huggingface dataset:
Automatic modular abstractions for template numerical constraints
Venue: arXiv cs.PL 2010
Year: 2010

We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floating-point variables and containing linear assignments and tests. In addition to loop-free code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. Our algorithms are based on new quantifier elimination and symbolic manipulation techniques. Given the specification of an abstract domain, and a program block, our method automatically outputs an implementation of the corresponding abstract transformer. It is thus a form of program transformation. The motivation of our work is data-flow synchronous programming languages, used for building control-command embedded systems, but it also applies to imperative and functional programming.
Sparse Signal Reconstruction via Iterative Support Detection
Venue: arXiv cs.IT 2010
Year: 2010

We present a novel sparse signal reconstruction method "ISD", aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical l_1 minimization approach. ISD addresses failed reconstructions of l_1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new reconstruction by solving the minimization problem \min{\sum_{i\not\in I}|x_i|:Ax=b}, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit (OMP) method, as well as its variants, because (i) the index set I in ISD is not necessarily nested or increasing and (ii) the minimization problem above updates all the components of x at the same time. We generalize the Null Space Property to Truncated Null Space Property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold--ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold--ISD has significant advantages over the classical l_1 minimization approach, as well as two state--of--the--art algorithms: the iterative reweighted l_1 minimization algorithm (IRL1) and the iterative reweighted least--squares algorithm (IRLS). MATLAB code is available for download from http://www.caam.rice.edu/~optimization/L1/ISD/.
Range Non-Overlapping Indexing
Venue: arXiv cs.DS 2010
Year: 2010

We study the non-overlapping indexing problem: Given a text T, preprocess it so that you can answer queries of the form: given a pattern P, report the maximal set of non-overlapping occurrences of P in T. A generalization of this problem is the range non-overlapping indexing where in addition we are given two indexes i,j to report the maximal set of non-overlapping occurrences between these two indexes. We suggest new solutions for these problems. For the non-overlapping problem our solution uses O(n) space with query time of O(m + occ_{NO}). For the range non-overlapping problem we propose a solution with O(n\log^εn) space for some 0<ε<1 and O(m + \log\log n + occ_{ij,NO}) query time.
Implementing Gröbner bases for operads
Venue: arXiv cs.SC 2010
Year: 2010

We present an implementation of the algorithm for computing Groebner bases for operads due to the first author and A. Khoroshkin. We discuss the actual algorithms, the choices made for the implementation platform and the data representation, and strengths and weaknesses of our approach.
A Simple Necessary and Sufficient Condition for the Double Unicast Problem
Venue: arXiv cs.IT 2010
Year: 2010

We consider a directed acyclic network where there are two source-terminal pairs and the terminals need to receive the symbols generated at the respective sources. Each source independently generates an i.i.d. random process over the same alphabet. Each edge in the network is error-free, delay-free, and can carry one symbol from the alphabet per use. We give a simple necessary and sufficient condition for being able to simultaneously satisfy the unicast requirements of the two source-terminal pairs at rate-pair $(1,1)$ using vector network coding. The condition is also sufficient for doing this using only "XOR" network coding and is much simpler compared to the necessary and sufficient conditions known from previous work. Our condition also yields a simple characterization of the capacity region of a double-unicast network which does not support the rate-pair $(1,1)$.
Exact Covers via Determinants
Venue: arXiv cs.DS 2010
Year: 2010

Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors polynomial in n and k. When we drop the partition constraint and permit arbitrary hyperedges of cardinality k, we obtain the exact cover by k-sets problem. We show it can be solved by a randomized polynomial space algorithm in time O*(c_k^n), where c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k, and follow from the new observation that Lovasz' perfect matching detection via determinants (1979) admits an embedding in the recently proposed inclusion-exclusion counting scheme for set covers, despite its inability to count the perfect matchings.
Regularization Techniques for Learning with Matrices
Venue: arXiv cs.LG 2010
Year: 2010

There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge). This work describes and analyzes a systematic method for constructing such matrix-based, regularization methods. In particular, we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate. Our methodology is based on the known duality fact: that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. We demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and kernel learning.
Covering Points by Disjoint Boxes with Outliers
Venue: arXiv cs.CG 2010
Year: 2010

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.
State of the cognitive interference channel: a new unified inner bound
Venue: arXiv cs.IT 2010
Year: 2010

The capacity region of the interference channel in which one transmitter non-causally knows the message of the other, termed the cognitive interference channel, has remained open since its inception in 2005. A number of subtly differing achievable rate regions and outer bounds have been derived, some of which are tight under specific conditions. In this work we present a new unified inner bound for the discrete memoryless cognitive interference channel. We show explicitly how it encompasses all known discrete memoryless achievable rate regions as special cases. The presented achievable region was recently used in deriving the capacity region of the general deterministic cognitive interference channel, and thus also the linear high-SNR deterministic approximation of the Gaussian cognitive interference channel. The high-SNR deterministic approximation was then used to obtain the capacity of the Gaussian cognitive interference channel to within 1.87 bits.
Faster Algorithms for Max-Product Message-Passing
Venue: arXiv cs.AI 2010
Year: 2010

Maximum A Posteriori inference in graphical models is often solved via message-passing algorithms, such as the junction-tree algorithm, or loopy belief-propagation. The exact solution to this problem is well known to be exponential in the size of the model's maximal cliques after it is triangulated, while approximate inference is typically exponential in the size of the model's factors. In this paper, we take advantage of the fact that many models have maximal cliques that are larger than their constituent factors, and also of the fact that many factors consist entirely of latent variables (i.e., they do not depend on an observation). This is a common case in a wide variety of applications, including grids, trees, and ring-structured models. In such cases, we are able to decrease the exponent of complexity for message-passing by 0.5 for both exact and approximate inference.
A Scalable VLSI Architecture for Soft-Input Soft-Output Depth-First Sphere Decoding
Venue: arXiv cs.AR 2010
Year: 2010

Multiple-input multiple-output (MIMO) wireless transmission imposes huge challenges on the design of efficient hardware architectures for iterative receivers. A major challenge is soft-input soft-output (SISO) MIMO demapping, often approached by sphere decoding (SD). In this paper, we introduce the - to our best knowledge - first VLSI architecture for SISO SD applying a single tree-search approach. Compared with a soft-output-only base architecture similar to the one proposed by Studer et al. in IEEE J-SAC 2008, the architectural modifications for soft input still allow a one-node-per-cycle execution. For a 4x4 16-QAM system, the area increases by 57% and the operating frequency degrades by 34% only.
Optimal bounds for sign-representing the intersection of two halfspaces by polynomials
Venue: arXiv cs.CC 2010
Year: 2010

The threshold degree of a function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p with f(x)=sgn p(x). We prove that the intersection of two halfspaces on {0,1}^n has threshold degree Omega(n), which matches the trivial upper bound and completely answers a question due to Klivans (2002). The best previous lower bound was Omega(sqrt n). Our result shows that the intersection of two halfspaces on {0,1}^n only admits a trivial 2^{Theta(n)}-time learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC learning DNF formulas and read-once Boolean formulas. The proof introduces a new technique of independent interest, based on Fourier analysis and matrix theory.
Putting Dots in Triangles
Venue: arXiv cs.DM 2010
Year: 2010

Given a right-angled triangle of squares in a grid whose horizontal and vertical sides are $n$ squares long, let N(n) denote the maximum number of dots that can be placed into the cells of the triangle such that each row, each column, and each diagonal parallel to the long side of the triangle contains at most one dot. It has been proven that $N(n) = \lfloor \frac{2n+1}{3} \rfloor$. In this note, we give a new proof of this result using linear programming techniques.
Fast algorithms for spherical harmonic expansions, III
Venue: arXiv cs.NA 2010
Year: 2010

We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on "Fast algorithms for spherical harmonic expansions." The requisite precomputations become manageable when organized as a "depth-first traversal" of the program's control-flow graph, rather than as the perhaps more natural "breadth-first traversal" that processes one-by-one each level of the multilevel procedure. We illustrate the results via several numerical examples.
Affine Grassmann Codes
Venue: arXiv cs.IT 2010
Year: 2010

We consider a new class of linear codes, called affine Grassmann codes. These can be viewed as a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. We determine the length, dimension, and the minimum distance of any affine Grassmann code. Moreover, we show that affine Grassmann codes have a large automorphism group and determine the number of minimum weight codewords.
On Improving the Representation of a Region Achieved by a Sensor Network
Venue: arXiv cs.NI 2010
Year: 2010

This report considers the class of applications of sensor networks in which each sensor node makes measurements, such as temperature or humidity, at the precise location of the node. Such spot-sensing applications approximate the physical condition of the entire region of interest by the measurements made at only the points where the sensor nodes are located. Given a certain density of nodes in a region, a more spatially uniform distribution of the nodes leads to a better approximation of the physical condition of the region. This report considers the error in this approximation and seeks to improve the quality of representation of the physical condition of the points in the region in the data collected by the sensor network. We develop two essential metrics which together allow a rigorous quantitative assessment of the quality of representation achieved: the average representation error and the unevenness of representation error, the latter based on a well-accepted measure of inequality used in economics. We present the rationale behind the use of these metrics and derive relevant theoretical bounds on them in the common scenario of a planar region of arbitrary shape covered by a sensor network deployment. A simple new heuristic algorithm is presented for each node to determine if and when it should sense or sleep to conserve energy while also preserving the quality of representation. Simulation results show that it achieves a significant improvement in the quality of representation compared to other related distributed algorithms. Interestingly, our results also show that improved spatial uniformity has the welcome side-effect of a significant increase in the network lifetime.
Fast Approximation in Subspaces by Doubling Metric Decomposition
Venue: arXiv cs.DS 2010
Year: 2010

In this paper we propose and study a new complexity model for approximation algorithms. The main motivation are practical problems over large data sets that need to be solved many times for different scenarios, e.g., many multicast trees that need to be constructed for different groups of users. In our model we allow a preprocessing phase, when some information of the input graph $G=(V,E)$ is stored in a limited size data structure. Next, the data structure enables processing queries of the form ``solve problem A for an input $S\subseteq V$''. We consider problems like {\sc Steiner Forest}, {\sc Facility Location}, {\sc $k$-Median}, {\sc $k$-Center} and {\sc TSP} in the case when the graph induces a doubling metric. Our main results are data structures of near-linear size that are able to answer queries in time close to linear in $|S|$. This improves over typical worst case reuniting time of approximation algorithms in the classical setting which is $Ω(|E|)$ independently of the query size. In most cases, our approximation guarantees are arbitrarily close to those in the classical setting. Additionally, we present the first fully dynamic algorithm for the Steiner tree problem.
Bifinite Chu Spaces
Venue: arXiv cs.LO 2010
Year: 2010

This paper studies colimits of sequences of finite Chu spaces and their ramifications. Besides generic Chu spaces, we consider extensional and biextensional variants. In the corresponding categories we first characterize the monics and then the existence (or the lack thereof) of the desired colimits. In each case, we provide a characterization of the finite objects in terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with respect to the monics of generic Chu spaces, and universal, homogeneous Chu spaces are shown to exist in this category. Unanticipated results driving this development include the fact that while for generic Chu spaces monics consist of an injective first and a surjective second component, in the extensional and biextensional cases the surjectivity requirement can be dropped. Furthermore, the desired colimits are only guaranteed to exist in the extensional case. Finally, not all finite Chu spaces (considered set-theoretically) are finite objects in their categories. This study opens up opportunities for further investigations into recursively defined Chu spaces, as well as constructive models of linear logic.
Enumerative Coding for Grassmannian Space
Venue: arXiv cs.IT 2010
Year: 2010

The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of $\F_q^n$. One enumerative coding method is based on a Ferrers diagram representation and on an order for $\Gr$ based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.
O-Minimal Hybrid Reachability Games
Venue: arXiv cs.LO 2010
Year: 2010

In this paper, we consider reachability games over general hybrid systems, and distinguish between two possible observation frameworks for those games: either the precise dynamics of the system is seen by the players (this is the perfect observation framework), or only the starting point and the delays are known by the players (this is the partial observation framework). In the first more classical framework, we show that time-abstract bisimulation is not adequate for solving this problem, although it is sufficient in the case of timed automata . That is why we consider an other equivalence, namely the suffix equivalence based on the encoding of trajectories through words. We show that this suffix equivalence is in general a correct abstraction for games. We apply this result to o-minimal hybrid systems, and get decidability and computability results in this framework. For the second framework which assumes a partial observation of the dynamics of the system, we propose another abstraction, called the superword encoding, which is suitable to solve the games under that assumption. In that framework, we also provide decidability and computability results.
Covering rough sets based on neighborhoods: An approach without using neighborhoods
Venue: arXiv cs.AI 2010
Year: 2010

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.
Codes on graphs: Duality and MacWilliams identities
Venue: arXiv cs.IT 2010
Year: 2010

A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier transform pair, whether or not the graphs have cycles. The original normal graph duality theorem follows as a corollary. Within this framework, MacWilliams identities are found for various local and global weight generating functions of general group or linear codes on graphs; this generalizes and provides a concise proof of the MacWilliams identity for linear time-invariant convolutional codes that was recently found by Gluesing-Luerssen and Schneider. Further MacWilliams identities are developed for terminated convolutional codes, particularly for tail-biting codes, similar to those studied recently by Bocharova, Hug, Johannesson and Kudryashov.
Mapping the spatiotemporal dynamics of calcium signaling in cellular neural networks using optical flow
Venue: arXiv cs.CE 2010
Year: 2010

An optical flow gradient algorithm was applied to spontaneously forming net- works of neurons and glia in culture imaged by fluorescence optical microscopy in order to map functional calcium signaling with single pixel resolution. Optical flow estimates the direction and speed of motion of objects in an image between subsequent frames in a recorded digital sequence of images (i.e. a movie). Computed vector field outputs by the algorithm were able to track the spatiotemporal dynamics of calcium signaling pat- terns. We begin by briefly reviewing the mathematics of the optical flow algorithm, and then describe how to solve for the displacement vectors and how to measure their reliability. We then compare computed flow vectors with manually estimated vectors for the progression of a calcium signal recorded from representative astrocyte cultures. Finally, we applied the algorithm to preparations of primary astrocytes and hippocampal neurons and to the rMC-1 Muller glial cell line in order to illustrate the capability of the algorithm for capturing different types of spatiotemporal calcium activity. We discuss the imaging requirements, parameter selection and threshold selection for reliable measurements, and offer perspectives on uses of the vector data.
Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories
Venue: arXiv cs.LO 2010
Year: 2010

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.