Boolean Network Learning in Vector Spaces for Genome-wide Network Analysis

Boolean networks (BNs) are one of the standard tools for modeling gene regulatory networks in biology but their learning has been limited to small networks due to computational difficulty. Aiming at unprecedented scalability, we focus on a subclass of BNs called AND/OR Boolean networks where Boolean formulas are restricted to a conjunction or a disjunction of literals. We represent an AND/OR BN with N nodes by an N x 2N binary matrix Q paired with an N dimensional integer vector theta called a threshold vector, a state of the BN by an N dimensional binary state vector s and a state transition by matrix operations on Q, theta and s. Given a list of state transitions S = s_0...s_L, we learn Q and theta in a continuous space by minimizing a cost function J(Q*,theta,S) w.r.t. a real number matrix Q* and theta while thresholding Q* into a binary matrix Q using theta so that Q represents an AND/OR BN realizing the target state transitions S. We conducted experiments with artificial and real data sets to check scalability and accuracy of our learning algorithm. First we randomly generated AND/OR BNs up to N=5,000 nodes and empirically confirmed O(N^2) learning time behavior using them. We also observed 99.8% bit-by-bit prediction accuracy (prediction accuracy = 1 - test error) with state transition data generated by AND/OR BNs. For real data, we learned genome-wide AND/OR BNs with 10,928 nodes for budding yeast from transcription profiling data sets, each containing 10,928 mRNAs and 40 transitions and achieved for instance 84.3% prediction accuracy and successfully extracted more than 6,000 small AND/ORs whose average prediction accuracy reaches much higher 94.9%.

Affine Extensions of Integer Vector Addition Systems with States

We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine $\mathbb{Z}$-VASS with the finite-monoid property (afmp-$\mathbb{Z}$-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-$\mathbb{Z}$-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-$\mathbb{Z}$-VASS reduces to reachability in a $\mathbb{Z}$-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-$\mathbb{Z}$-VASS are semilinear, and in particular enables us to show that reachability in $\mathbb{Z}$-VASS with transfers and $\mathbb{Z}$-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine $\mathbb{Z}$-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine $\mathbb{Z}$-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine $\mathbb{Z}$-VASS with monogenic matrix monoid and undecidable reachability relation.

Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

Ext-LOUDS: A Space Efficient Extended LOUDS Index for Superset Query

Superset query is widely used in object-oriented databases, data mining, and many other fields. Trie is an efficient index for superset query, whereas most existing trie index aim at improving query performance while ignoring storage overheads. To solve this problem, in this paper, we propose an efficient extended Level-Ordered Unary Degree Sequence (LOUDS) index: Ext-LOUDS. Ext-LOUDS expresses a trie by 1 integer vector and 3 bit vectors directly map each NodeID to its corresponding position, thus accelerating some key operations needed for superset query. Based on Ext-LOUDS, an efficient superset query algorithm, ELOUDS-Super, is designed. Experimental results on both real and synthetic datasets show that Ext-LOUDS can decrease 50%–60% space overheads compared with trie while maintaining a relative good query performance.

On the Performance Analysis for CSIDH-Based Cryptosystems

In this paper, we present the performance and security analysis for various commutative SIDH (CSIDH)-based algorithms. As CSIDH offers a smaller key size than SIDH and provides a relatively efficient signature scheme, numerous CSIDH-based key exchange algorithms have been proposed to optimize the CSIDH. In CSIDH, the private key is an ideal class in a class group, which can be represented by an integer vector. As the number of ideal classes represented by these vectors determines the security level of CSIDH, it is important to analyze whether the different vectors induce the same public key. In this regard, we generalize the existence of a collision for a base prime p≡7mod8. Based on our result, we present a new interval for the private key to have a similar security level for the various CSIDH-based algorithms for a fair comparison of the performance. Deduced from the implementation result, we conclude that for a prime p≡7mod8, CSIDH on the surface using the Montgomery curves is the most likely to be efficient. For a prime p≡3mod8, CSIDH on the floor using the hybrid method with Onuki’s collision-free method is the most likely to be efficient and secure.

Solving a Modified Syndrome Decoding Problem using Integer Programming

In this article, we model a variant of the well-known syndrome decoding problem as a linear optimization problem. Most common algorithms used for solving optimization problems, e.g. the simplex algorithm, fail to find a valid solution for the syndrome decoding problem over a finite field. However, our simulations prove that a slightly modified version of the syndrome decoding problem can be solved by the simplex algorithm. More precisely, the algorithm returns a valid error vector when the syndrome vector is an integer vector, i.e.,the matrix-vector multiplication, is realized over Z, instead of Fq.

Inhomogeneous Partition Regularity

We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved that the system $Ax = b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general (commutative) ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new 'direct' proof of Rado’s result.

Abstract Tropical Linear Programming

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. We obtain an algorithm based on an axiomatic approach to this generalization. It builds on the introduction of signed tropical matroids based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information. Finally, we establish an upper bound for our feasibility algorithm applied to a system of min-plus-inequalities in terms of the secondary fan of a product of two simplices. The appropriate complexity measure is a shortest integer vector in a cone of the secondary fan associated to the system.

The Smith normal form distribution of a random integer matrix

International audience We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.