Guoyin Li

Hwa Hui Tew, Fan Ding, Gaoxuan Li et al.

Higher-order sensor networks are more accurate in characterizing the nonlinear dynamics of sensory time-series data in modern industrial settings by allowing multi-node connections beyond simple pairwise graph edges. In light of this, we propose a deep spatio-temporal hypergraph convolutional neural network for soft sensing (ST-HCSS). In particular, our proposed framework is able to construct and leverage a higher-order graph (hypergraph) to model the complex multi-interactions between sensor nodes in the absence of prior structural knowledge. To capture rich spatio-temporal relationships underlying sensor data, our proposed ST-HCSS incorporates stacked gated temporal and hypergraph convolution layers to effectively aggregate and update hypergraph information across time and nodes. Our results validate the superiority of ST-HCSS compared to existing state-of-the-art soft sensors, and demonstrates that the learned hypergraph feature representations aligns well with the sensor data correlations. The code is available at https://github.com/htew0001/ST-HCSS.git

Peiran Yu, Guoyin Li, Ting Kei Pong

Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of $\frac12$ for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we show that the KL exponent is $\frac12$ for several important convex optimization models, including some semidefinite-programming-representable functions and some functions that involve $C^2$-cone reducible structures, under conditions such as strict complementarity. Our results are applicable to concrete optimization models such as group fused Lasso and overlapping group Lasso. In addition, for nonconvex models, we show that the KL exponent of many difference-of-convex functions can be derived from that of their natural majorant functions, and the KL exponent of the Bregman envelope of a function is the same as that of the function itself. Finally, we estimate the KL exponent of the sum of the least squares function and the indicator function of the set of matrices of rank at most $k$.

Guoyin Li, Ting Kei Pong

We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method to minimize the sum of a proper closed function $g$ and a smooth function $f$ with a Lipschitz continuous gradient, we show that if the step-size parameter is smaller than a computable threshold and the sequence generated has a cluster point, then it gives a stationary point of the optimization problem. Convergence of the whole sequence and a local convergence rate are also established under the additional assumption that $f$ and $g$ are semi-algebraic. We also give simple sufficient conditions guaranteeing the boundedness of the sequence generated. We then apply our nonconvex DR splitting method to finding a point in the intersection of a closed convex set $C$ and a general closed set $D$ by minimizing the squared distance to $C$ subject to $D$. We show that if either set is bounded and the step-size parameter is smaller than a computable threshold, then the sequence generated from the DR splitting method is actually bounded. Consequently, the sequence generated will have cluster points that are stationary for an optimization problem, and the whole sequence is convergent under an additional assumption that $C$ and $D$ are semi-algebraic. We achieve these results based on a new merit function constructed particularly for the DR splitting method. Our preliminary numerical results indicate that our DR splitting method usually outperforms the alternating projection method in finding a sparse solution of a linear system, in terms of both the solution quality and the number of iterations taken.