Tianran Chen

Tianran Chen, Dhagash Mehta

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions.Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.

Tianran Chen, Jakub Marecek, Dhagash Mehta et al.

We compare three formulations of stationary equations of the Kuramoto model as systems of polynomial equations. In the comparison, we present bounds on the numbers of real equilibria based on the work of Bernstein, Kushnirenko, and Khovanskii, and performance of methods for the optimisation over the set of equilibria based on the work of Lasserre, both of which could be of independent interest.

Baoquan Zhang, Zhehao Yu, Lisai Zhang et al.

Parameter Efficient Fine-Tuning (PEFT) is a key technique for adapting a large pretrained model to downstream tasks by fine-tuning only a small number of parameters. Recent methods based on Fourier transforms have further reduced the fine-tuned parameters scale by only fine-tuning a few spectral coefficients. Its basic assumption is that the weight change δW is a spatial-domain matrix with a sparse spectrum. However, in this paper, we observe that the spectrum of weight change is not sparse, but instead distributed like power-uniform. This fact implies that fine-tuning only a few spectral coefficients is insufficient to accurately model the weight change with uniform spectrum. To address this issue, we propose to seek an invertible transformation that can transform a latent spatial-domain matrix with sparse spectrum to the weight change, and then perform PEFT on such sparse spectrum domain with few spectral coefficients, called S2FT. To seek such transformation, we first pre-estimate a coarse weight change as a prior. Then, inspired by that sparse spectrum often correspond to locally smooth spatial structures, we regard this transformation as a row and column rearrangement operation on the pre-estimated weight change that smooth spatial structures while keep the structure information of neurons. Finally, we propose to solve the rearrangement search problem in a simple nearest neighbor search manner, thereby obtaining the invertible transformation. Extensive results show our S2FT achieves superior performance by only using 0.08% training parameters.

Dhagash Mehta, Tianran Chen, Tingting Tang et al.

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized $L_2$ regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Using these upper bounds and utilizing a numerical algebraic geometry method, we find all stationary points of modest depth and matrix size. We show that in the presence of the non-zero regularization, deep linear networks indeed possess local minima which are not the global minima. Our computational results clarify certain aspects of the loss surfaces of deep linear networks and provide novel insights.

Christian Knoll, Franz Pernkopf, Dhagash Mehta et al.

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.