Chang Liu

Arnaud Deza, Chang Liu, Pashootan Vaezipoor et al. · utoronto

It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for multiplying two $2\times 2$ matrices in 7 instead of 8 multiplications. This gives rise to the constraint satisfaction problem of fast matrix multiplication, where a set of $R < NMP$ multiplication terms must be chosen and combined such that they satisfy correctness constraints on the output matrix. Despite its highly combinatorial nature, this problem has not been exhaustively examined from that perspective, as evidenced for example by the recent deep reinforcement learning approach of AlphaTensor. In this work, we propose a simple yet novel Constraint Programming approach to find non-commutative algorithms for fast matrix multiplication or provide proof of infeasibility otherwise. We propose a set of symmetry-breaking constraints and valid inequalities that are particularly helpful in proving infeasibility. On the feasible side, we find that exploiting solver performance variability in conjunction with a sparsity-based problem decomposition enables finding solutions for larger (feasible) instances of fast matrix multiplication. Our experimental results using CP Optimizer demonstrate that we can find fast matrix multiplication algorithms for matrices up to $3\times 3$ in a short amount of time.

Qi Zhang, Chang Liu, Stephen Wu et al.

In the last few years, de novo molecular design using machine learning has made great technical progress but its practical deployment has not been as successful. This is mostly owing to the cost and technical difficulty of synthesizing such computationally designed molecules. To overcome such barriers, various methods for synthetic route design using deep neural networks have been studied intensively in recent years. However, little progress has been made in designing molecules and their synthetic routes simultaneously. Here, we formulate the problem of simultaneously designing molecules with the desired set of properties and their synthetic routes within the framework of Bayesian inference. The design variables consist of a set of reactants in a reaction network and its network topology. The design space is extremely large because it consists of all combinations of purchasable reactants, often in the order of millions or more. In addition, the designed reaction networks can adopt any topology beyond simple multistep linear reaction routes. To solve this hard combinatorial problem, we present a powerful sequential Monte Carlo algorithm that recursively designs a synthetic reaction network by sequentially building up single-step reactions. In a case study of designing drug-like molecules based on commercially available compounds, compared with heuristic combinatorial search methods, the proposed method shows overwhelming performance in terms of computational efficiency and coverage and novelty with respect to existing compounds.

Jinhua Zhu, Yingce Xia, Chang Liu et al.

Molecular conformation generation aims to generate three-dimensional coordinates of all the atoms in a molecule and is an important task in bioinformatics and pharmacology. Previous methods usually first predict the interatomic distances, the gradients of interatomic distances or the local structures (e.g., torsion angles) of a molecule, and then reconstruct its 3D conformation. How to directly generate the conformation without the above intermediate values is not fully explored. In this work, we propose a method that directly predicts the coordinates of atoms: (1) the loss function is invariant to roto-translation of coordinates and permutation of symmetric atoms; (2) the newly proposed model adaptively aggregates the bond and atom information and iteratively refines the coordinates of the generated conformation. Our method achieves the best results on GEOM-QM9 and GEOM-Drugs datasets. Further analysis shows that our generated conformations have closer properties (e.g., HOMO-LUMO gap) with the groundtruth conformations. In addition, our method improves molecular docking by providing better initial conformations. All the results demonstrate the effectiveness of our method and the great potential of the direct approach. The code is released at https://github.com/DirectMolecularConfGen/DMCG

Jiaxin Shi, Chang Liu, Lester Mackey

We introduce a new family of particle evolution samplers suitable for constrained domains and non-Euclidean geometries. Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent minimize the Kullback-Leibler (KL) divergence to constrained target distributions by evolving particles in a dual space defined by a mirror map. Stein Variational Natural Gradient exploits non-Euclidean geometry to more efficiently minimize the KL divergence to unconstrained targets. We derive these samplers from a new class of mirrored Stein operators and adaptive kernels developed in this work. We demonstrate that these new samplers yield accurate approximations to distributions on the simplex, deliver valid confidence intervals in post-selection inference, and converge more rapidly than prior methods in large-scale unconstrained posterior inference. Finally, we establish the convergence of our new procedures under verifiable conditions on the target distribution.