Monotone Network Research Articles

Size and Depth of Monotone Neural Networks: Interpolation and Approximation.

We study monotone neural networks with threshold gates where all the weights (other than the biases) are nonnegative. We focus on the expressive power and efficiency of the representation of such networks. Our first result establishes that every monotone function over [0,1]d can be approximated within arbitrarily small additive error by a depth-4 monotone network. When , we improve upon the previous best-known construction, which has a depth of d+1 . Our proof goes by solving the monotone interpolation problem for monotone datasets using a depth-4 monotone threshold network. In our second main result, we compare size bounds between monotone and arbitrary neural networks with threshold gates. We find that there are monotone real functions that can be computed efficiently by networks with no restriction on the gates, whereas monotone networks approximating these functions need exponential size in the dimension.

Distributed Identification of Contracting and/or Monotone Network Dynamics

This article proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting—a strong form of nonlinear stability—and/or monotone, i.e., order relations between states are preserved. The main challenges that we address are simultaneously searching for model parameters and a certificate of model stability, and scalability to networks with hundreds or thousands of nodes. We propose a model set that admits convex constraints for stability and monotonicity, and both model and stability certificates have a separable structure that allows distributed identification via the alternating directions method of multipliers. The performance and scalability of the approach is illustrated on a variety of linear and nonlinear case studies, including a nonlinear traffic network with a 200-D state space.

Reprogramming multistable monotone systems with application to cell fate control.

Multistability is a key property of dynamical systems modeling cellular regulatory networks implicated in cell fate decisions, where, different stable steady states usually represent distinct cell phenotypes. Monotone network motifs are highly represented in these regulatory networks. In this paper, we leverage the properties of monotone dynamical systems to provide theoretical results that guide the selection of inputs that trigger a transition, i.e., reprogram the network, to a desired stable steady state. We first show that monotone dynamical systems with bounded trajectories admit a minimum and a maximum stable steady state. Then, we provide input choices that are guaranteed to reprogram the system to these extreme steady states. For intermediate states, we provide an input space that is guaranteed to contain an input that reprograms the system to the desired state. We then provide implementation guidelines for finite-time procedures that search this space for such an input, along with rules to prune parts of the space during search. We demonstrate these results on simulations of two recurrent regulatory network motifs: self-activation within mutual antagonism and self-activation within mutual cooperation. Our results depend uniquely on the structure of the network and are independent of specific parameter values.

Fixing monotone Boolean networks asynchronously

The asynchronous automaton associated with a Boolean network f:{0,1}n→{0,1}n is considered in many applications. It is the finite deterministic automaton with set of states {0,1}n, alphabet {1,…,n}, where the action of letter i on a state x consists in switching the ith component if fi(x)≠xi or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word w fixes f if, for all states x, the result of the action of w on x is a fixed point of f. In this paper, we ask for the existence of fixing words, and their minimal length. Firstly, our main results concern the minimal length of words that fix monotone networks. We prove that there exists a monotone network f with n components such that any word fixing f has length Ω(n2). Conversely, we construct a word of length O(n3) that fixes all monotone networks with n components. Secondly, we refine and extend our results to different classes of networks.

Convergence Theory for Unified Discrete-Time RNNs with Quasi-Symmetric Connection

This paper presents the global convergence theory of the discrete-time uniform pseudo projection anti-monotone network with the quasi–symmetric matrix, which removes the connection matrix constraints. The theory widens the range of applications of the discrete–time uniform pseudo projection anti–monotone network and is valid for many kinds of discrete recurrent neural network models.

Spanning network games

We study fundamental properties of monotone network enterprises which contain public vertices and have positive and negative costs on edges and vertices. Among the properties studied are the nonemptiness of the core, characterization of nonredundant core constraints, ease of computation of the core and the nucleolus, and cases of decomposition of the core and the nucleolus.

The complexity of central slice functions

Let f be a monotone Boolean function over X = { x 1,…, x n }. The k-slice of f is the function f k = ( f^ T k n ) vT k+1 n , where T k n is the kth threshold function. Berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of k k imply lower bounds of the same order on the combinational complexity of f, and that if f has large combinational complexity, then some slice of f must have large monotone complexity. However, this latter result does not specify any particular slice and it is known that some (nontrivial) slice functions of NP-complete predicates have linear complexity. In this paper we consider the 1 2 n slice functions and show that, for three basic monotone Boolean NP-complete functions, this slice is also NP-complete. In addition the 1 2 n- slice of some Boolean matrix functions F is studied, and is proved to be no easier to compute than F.

A 2.5n lower bound on the monotone network complexity of T3n

The k-th threshold function, T k n , is defined as: $$T_k^n \left( {x_1 ,...,x_n } \right) = \left\{ \begin{gathered} 1 if \sum\limits_{i = 1}^n {x_i \geqq k} \hfill \\ 0 otherwise \hfill \\ \end{gathered} \right.$$ where x ie{0,1} and the summation is arithmetic. We prove that any monotone network computing T 3/n(x 1,...,x n) contains at least 2.5n-5.5 gates.

An [formula omitted] lower bound on the monotone network complexity of the nth degree convolution

We prove an Ω(n 4 3 ) lower bound on the number of Λ-gates in any monotone network computing the n th degree covolution.

A 4n-lower bound on the monotone network complexity of a one-output boolean function

The main purpose of Boolean network theory is to find functions f:{0, 1} n → {0, 1} with large network complexity. The best known lower bound over the complete basis of all binary functions is of size 3n for a non-monotone function (Blum (1982)). Bloniarz (1979) has proved a 3n-lower bound for the majority-function over the monotone basis. In this paper a special function is presented for which a lower bound of size 4n over the monotone basis can be proved.

An n3/2 lower bound on the monotone network complexity of the Boolean convolution

In this paper, a general lower bound on the monotone network complexity of semidisjoint bilinear forms is proved. By this method an n 3/2 lower bound for the Boolean convolution is obtained. Up to now the best known lower bound for the Boolean convolution was of size n 4/3 ( Blum 1981 , IEEE Annual Sympos. Found. Comput. Sci. 22 , 101–108).

A new lower bound on the monotone network complexity of Boolean sums

Neciporuk [3], Lamagna/Savage [1] and Tarjan [6] determined the monotone network complexity of a set of Boolean sums if each two sums have at most one variable in common. By this result they could define explicitely a set of n Boolean sums which depend on n variables and whose monotone complexity is of order n 3/2. In the main theorem of this paper we prove a more general lower bound on the monotone network complexity of Boolean sums. Our lower bound is for many Boolean sums the first nontrivial lower bound. On the other side we can prove that the best lower bound which the main theorem yields is the n 3/2-bound cited above. For the proof we use the technical trick of assuming that certain functions are given for free.

Some remarks on Boolean sums

Neciporuk, Lamagna/Savage and Tarjan determined the monotone network complexity of a set of Boolean sums if any two sums have at most one variable in common. Wegener then solved the case that any two sums have at most k variables in common. We extend his methods and results and consider the case that any set of h +1 distinct sums have at most k variables in common. We use our general results to explicitly construct a set of n Boolean sums over n variables whose monotone complexity is of order n 5/3. The best previously known bound was of order n 3/2. Related results were obtained independently by Pippenger.