The RNA Newton polytope and learnability of energy parameters

Abstract

Motivation: Computational RNA structure prediction is a mature important problem that has received a new wave of attention with the discovery of regulatory non-coding RNAs and the advent of high-throughput transcriptome sequencing. Despite nearly two score years of research on RNA secondary structure and RNA–RNA interaction prediction, the accuracy of the state-of-the-art algorithms are still far from satisfactory. So far, researchers have proposed increasingly complex energy models and improved parameter estimation methods, experimental and/or computational, in anticipation of endowing their methods with enough power to solve the problem. The output has disappointingly been only modest improvements, not matching the expectations. Even recent massively featured machine learning approaches were not able to break the barrier. Why is that?Approach: The first step toward high-accuracy structure prediction is to pick an energy model that is inherently capable of predicting each and every one of known structures to date. In this article, we introduce the notion of learnability of the parameters of an energy model as a measure of such an inherent capability. We say that the parameters of an energy model are learnable iff there exists at least one set of such parameters that renders every known RNA structure to date the minimum free energy structure. We derive a necessary condition for the learnability and give a dynamic programming algorithm to assess it. Our algorithm computes the convex hull of the feature vectors of all feasible structures in the ensemble of a given input sequence. Interestingly, that convex hull coincides with the Newton polytope of the partition function as a polynomial in energy parameters. To the best of our knowledge, this is the first approach toward computing the RNA Newton polytope and a systematic assessment of the inherent capabilities of an energy model. The worst case complexity of our algorithm is exponential in the number of features. However, dimensionality reduction techniques can provide approximate solutions to avoid the curse of dimensionality.Results: We demonstrated the application of our theory to a simple energy model consisting of a weighted count of A-U, C-G and G-U base pairs. Our results show that this simple energy model satisfies the necessary condition for more than half of the input unpseudoknotted sequence–structure pairs (55%) chosen from the RNA STRAND v2.0 database and severely violates the condition for ∼13%, which provide a set of hard cases that require further investigation. From 1350 RNA strands, the observed 3D feature vector for 749 strands is on the surface of the computed polytope. For 289 RNA strands, the observed feature vector is not on the boundary of the polytope but its distance from the boundary is not more than one. A distance of one essentially means one base pair difference between the observed structure and the closest point on the boundary of the polytope, which need not be the feature vector of a structure. For 171 sequences, this distance is l...

Exact Learning of RNA Energy Parameters From Structure

“…It turns out that only 126 sequences out of the initial 2277 remain in A ′ . Note that our condition here is more stringent than the necessary condition in [1], and that is why fewer sequences satisfy this condition. After 100 iterations (MaxIterations = 100) which took less than a minute, the algorithm returned 3 polytopes that are compatible, i.e.…”

“…The question that we asked before [1] was: does there exist nonzero parameters h † such that for every (x, y) ∈ D, y = arg min s G(x, s, h † )? We ask a slightly relaxed version of that question in this paper: does there exist nonzero parameters h † such that for every (x, y) ∈ D, G(x, y, h † ) = min s G(x, s, h † )?…”

“…Let (x, y) ∈ D and 0 = h † ∈ R k . We previously showed in [1] that if y minimizes G(x, s, h † ) as a function of s, then c(x, y) ∈ ∂N (x), i.e. the feature vector of (x, y) is on the boundary of the Newton polytope of x.…”

“…The first step towards high accuracy RNA structure prediction is to make sure that the energy model is inherently capable of predicting every observed structure, but it is not over-capable, as accurate estimation of the parameters of a highly complex model often requires a myriad of experimental data, and lack of sufficient experimental data causes overfitting. Recently, we gave a systematic method to assess that inherent capability of a given energy model [1]. Our algorithm decides whether the parameters of an energy model are learnable.…”

Exact Learning of RNA Energy Parameters from Structure

“…A necessary and sufficient condition for learnability of parameters is derived, which is based on computing the convex hull of union of translated Newton polytopes of input sequences [1]. The set of learned energy parameters is characterized as the convex cone generated by the normal vectors to those facets of the resulting polytope that are incident to the origin.…”