Irreversible Markov chains in spin models: Topological excitations

We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Frechet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent z \sim 2 at the critical temperature to z \sim 0 in the limit of zero temperature. We confirm the event-chain algorithm's fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.Comment: 5 pages, 5 figure

Mixing and perfect sampling in one-dimensional particle systems

We study the approach to equilibrium of the event-chain Monte Carlo (ECMC) algorithm for the one-dimensional hard-sphere model. Using the connection to the coupon-collector problem, we prove that a specific version of this local irreversible Markov chain realizes perfect sampling in O(N^2 log N) events, whereas the reversible local Metropolis algorithm requires O(N^3 log N) time steps for mixing. This confirms a special case of an earlier conjecture about O(N^2 log N) scaling of mixing times of ECMC and of the forward Metropolis algorithm, its discretized variant. We furthermore prove that sequential ECMC (with swaps) realizes perfect sampling in O(N^2) events. Numerical simulations indicate a cross-over towards O(N^2 log N) mixing for the sequential forward swap Metropolis algorithm, that we introduce here. We point out open mathematical questions and possible applications of our findings to higher-dimensional statistical-physics models.Comment: 7 pages, 7 figure

Event-chain Monte Carlo with factor fields

International audienceWe study the dynamics of one-dimensional (1D) interacting particles simulated with the event-chain Monte Carlo algorithm (ECMC). We argue that previous versions of the algorithm suffer from a mismatch in the factor potential between different particle pairs (factors) and show that in 1D models, this mismatch is overcome by factor fields. ECMC with factor fields is motivated, in 1D, for the harmonic model, and validated for the Lennard-Jones model as well as for hard spheres. In 1D particle systems with short-range interactions, autocorrelation times generally scale with the second power of the system size for reversible Monte Carlo dynamics, and with its first power for regular ECMC and for molecular-dynamics. We show, using numerical simulations, that they grow only with the square root of the systems size for ECMC with factor fields. Mixing times, which bound the time to reach equilibrium from an arbitrary initial configuration, grow with the first power of the system size

Theoretical Explanation of Activation Sparsity through Flat Minima and Adversarial Robustness

A recent empirical observation of activation sparsity in MLP layers offers an opportunity to drastically reduce computation costs for free. Despite several works attributing it to training dynamics, the theoretical explanation of activation sparsity's emergence is restricted to shallow networks, small training steps well as modified training, even though the sparsity has been found in deep models trained by vanilla protocols for large steps. To fill the three gaps, we propose the notion of gradient sparsity as the source of activation sparsity and a theoretical explanation based on it that explains gradient sparsity and then activation sparsity as necessary steps to adversarial robustness w.r.t. hidden features and parameters, which is approximately the flatness of minima for well-learned models. The theory applies to standardly trained LayerNorm-ed pure MLPs, and further to Transformers or other architectures if noises are added to weights during training. To eliminate other sources of flatness when arguing sparsities' necessity, we discover the phenomenon of spectral concentration, i.e., the ratio between the largest and the smallest non-zero singular values of weight matrices is small. We utilize random matrix theory (RMT) as a powerful theoretical tool to analyze stochastic gradient noises and discuss the emergence of spectral concentration. With these insights, we propose two plug-and-play modules for both training from scratch and sparsity finetuning, as well as one radical modification that only applies to from-scratch training. Another under-testing module for both sparsity and flatness is also immediate from our theories. Validational experiments are conducted to verify our explanation. Experiments for productivity demonstrate modifications' improvement in sparsity, indicating further theoretical cost reduction in both training and inference

Research of growth mechanism of ceramic coatings fabricated by micro-arc oxidation on magnesium alloys at high current mode

AbstractMicro-arc oxidation (MAO) coatings of ZK60 magnesium alloys were formed in a self-developed dual electrolyte composed of sodium silicate and phosphate at the high constant current of 1.8 A (15 A/dm2). The MAO process and growth mechanism were investigated by scanning electron microscopy (SEM) coupled with an energy dispersive spectrometer (EDS), confocal laser scanning microscopy and X-ray diffraction (XRD). The results indicate that the growth process of MAO coating mainly goes through “forming → puncturing → rapid growth of micro-arc oxidation →large arc discharge → self-repairing”. The coating grows inward and outward at the same time in the initial stage, but outward growth of the coating is dominant later. Mg, Mg2SiO4 and MgO are the main phases of ceramic coating

A compact robotic device for upper-limb reaching rehabilitation

This paper presents a compact linear-motion robotic device for upper-extremity reaching rehabilitation. Starting from conceptual design, the paper describes electronic circuit design and program development. The work develops a prototype that provides active and passive rehabilitation training. In active training, subjects actively move their arm with assistive or resistive force from the device to finish predefined displacement and force profiles. In passive training, subjects remain passive while the device moves the limb following the pre-defined displacement profile. Engineering specifications with adequate safety factor are determined and standard electronic and readily available mechanical components are exploited to keep the total cost low