Random Instantiations Research Articles

Optimizing ADMM and Over-Relaxed ADMM Parameters for Linear Quadratic Problems

The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate of ADMM. However, determining optimal algorithmic parameters, including both the associated penalty and relaxation parameters, often relies on empirical approaches tailored to specific problem domains and contextual scenarios. Incorrect parameter selection can significantly hinder ADMM's convergence rate. To address this challenge, in this paper we first propose a general approach to optimize the value of penalty parameter, followed by a novel closed-form formula to compute the optimal relaxation parameter in the context of linear quadratic problems (LQPs). We then experimentally validate our parameter selection methods through random instantiations and diverse imaging applications, encompassing diffeomorphic image registration, image deblurring, and MRI reconstruction.

Predicting Defects in Soft Sphere Packings near Jamming Using the Force Network Ensemble.

Amorphous systems of soft particles above jamming have more contacts than are needed to achieve mechanical equilibrium. The force network of a granular system with a fixed contact network is thus underdetermined and can be characterized as a random instantiation within the space of the force network ensemble. In this Letter, we show that defect contacts that are not necessary for stability of the system can be uniquely identified by examining the boundaries of this space of allowed force networks. We further show that, for simulations in the near jamming limit, this identification is nearly always correct and that defect contacts are broken under decompression of the system.

A Machine Learning-based surrogate modeling framework for predicting the history-dependent deformation of dual phase microstructures

A Machine Learning (ML)-based surrogate modeling framework is developed to predict the heterogeneous deformation behavior of dual phase microstructures. The deformation is first simulated using a dislocation density-based J2 plasticity Finite Element (FE) model, whose results form the basis for surrogate model training and validation. Long Short Term Memory (LSTM)-based ML models, with different architectures, are employed to predict the spatio-temporal evolution of three output variables: effective strain, von Mises effective stress, and the stress triaxiality ratio. Two metrics, the mean average error (MAE) and the coefficient of determination, R2, are used to assess the performance of the models and different architectures. Based on our analysis, the LSTM model is generally found to predict the spatio-temporal deformation fields with reasonable accuracy, even for untrained microstructures with varying microstructural attributes and random instantiations. The LSTM model is also used to predict aggregate properties, such as the stress–strain response and the strain partitioning in the dual phase microstructures.

On the Minimax Misclassification Ratio of Hypergraph Community Detection

Community detection in hypergraphs is explored. Under a generative hypergraph model called “ $d$ -wise hypergraph stochastic block model” ( $d$ - $\mathtt {hSBM}$ ), which naturally extends the stochastic block model ( $\mathtt {SBM}$ ) from graphs to $d$ -uniform hypergraphs, the fundamental limit of the misclassification ratio (the loss function in the community detection problem) is studied. For the converse part, a lower bound of the minimax risk, that is, the minimax expected misclassification ratio, is derived. Asymptotically, it decays exponentially fast to zero as the number of nodes tends to infinity, and the rate function is a weighted combination of several divergence terms, each of which is the Renyi divergence of order 1/2 between two Bernoulli distributions. The Bernoulli distributions involved in the characterization of the rate function are those governing the random instantiation of hyperedges in $d$ - $\mathtt {hSBM}$ . For the achievability part, we propose a two-step polynomial-time algorithm which, with high probability, has a misclassification ratio with a decaying exponent that is asymptotically greater than or equal to that of our proposed lower bound. The first step of the algorithm is a hypergraph spectral clustering method, which achieves partial recovery to a certain precision level. The second step is a local refinement method, which leverages the underlying probabilistic model along with parameter estimation from the outcome of the first step. To characterize the asymptotic performance of the proposed algorithm, we first derive a sufficient condition for attaining weak consistency in the hypergraph spectral clustering step. Then, under the guarantee of weak consistency in the first step, we upper bound the loss (with high probability) attained in the local refinement step by an exponentially decaying function of the size of the hypergraph and characterize the decaying rate. Compared to existing works in $\mathtt {SBM}$ , the main technical challenge lies in the complex structure of error events since community relations become much more complicated. The experimental results on both the synthetic data and real-world datasets validate our theoretical finding that the refinement step is critical in achieving the optimal statistical limit. For the special case $d=3$ , it is further shown by analyzing the performance of MLE that the proposed lower bound of the minimax risk is tight, and hence, the exponential decaying rate of the asymptotic minimax risk is characterized. We conjecture that this continues to hold for general $d$ as well.

Monte-Carlo Acceleration of Bilateral Filter and Non-Local Means.

We propose stochastic bilateral filter (SBF) and stochastic non-local means (SNLM), efficient randomized processes that agree with conventional bilateral filter (BF) and non-local means (NLM) on average, respectively. By Monte-Carlo, we repeat this process a few times with different random instantiations so that they can be averaged to attain the correct BF/NLM output. The computational bottleneck of the SBF and SNLM are constant with respect to the window size and the color dimension of the edge image, meaning the execution times for color and hyperspectral images are nearly the same as for the grayscale images. In addition, for SNLM, the complexity is constant with respect to the block size. The proposed stochastic filter implementations are considerably faster than the conventional and existing "fast" implementations for high dimensional image data.

On the adaptivity gap of stochastic orienteering

The input to the stochastic orienteering problem (Gupta et al. in SODA, pp 1522---1538, 2012) consists of a budget B and metric (V, d) where each vertex $$v\in V$$v?V has a job with a deterministic reward and a random processing time (drawn from a known distribution). The processing times are independent across vertices. The goal is to obtain a non-anticipatory policy (originating from a given root vertex) to run jobs at different vertices, that maximizes expected reward, subject to the total distance traveled plus processing times being at most B. An adaptive policy can choose the next vertex to visit based on observed random instantiations. Whereas, a non-adaptive policy is just given by a fixed ordering of vertices. The adaptivity gap is the worst-case ratio of the optimal adaptive and non-adaptive rewards. We prove an $$\varOmega \left[ (\log \log B)^{1/2}\right] $$Ω(loglogB)1/2 lower bound on the adaptivity gap of stochastic orienteering. This provides a negative answer to the O(1)-adaptivity gap conjectured by Gupta et al. (2012), and comes close to the $$O(\log \log B)$$O(loglogB) upper bound. This result holds even on a line metric. We also show an $$O(\log \log B)$$O(loglogB) upper bound on the adaptivity gap for the correlated stochastic orienteering problem, where the reward of each job is random and possibly correlated to its processing time. Using this, we obtain an improved quasi-polynomial time $$ \min \{\log n,\log B\}\cdot \tilde{O}(\log ^2\log B)$$min{logn,logB}·O~(log2logB)-approximation algorithm for correlated stochastic orienteering.

Statistical Prediction of “Reasonable Worst-Case” Crosstalk in Cable Bundles

Worst-case estimates of crosstalk in cable bundles are useful for flagging potential problems, but may also flag problems that only occur very rarely, due to the random variation of wire positions and other characteristics of the harness. Prediction of crosstalk that may realistically occur requires statistical methods. Monte Carlo simulation techniques are often used to account for statistical variation, but are time consuming and do not provide intuition toward the cause of, or solution to, problems. Here, we investigate prediction of the statistically ldquoreasonable worst-caserdquo crosstalk by forming probability distributions using inductance and capacitance parameters from a single harness cross section and using lumped-element approximations for crosstalk that account for strong coupling within the harness when the circuit is electrically small. The accuracy of this technique was evaluated through comparison to simulation results using the random displacement spline interpolation method for multiple random instantiations of several harness configurations. Tests were performed while varying the size of the bundle, its height above the return plane, the value of load impedances, and the presence of a return wire. The reasonable worst-case crosstalk was estimated within about 5 dB or less in each case.

Towards a realistic biomechanical model of the thumb: the choice of kinematic description may be more critical than the solution method or the variability/uncertainty of musculoskeletal parameters

A biomechanical model of the thumb can help researchers and clinicians understand the clinical problem of how anatomical variability contributes to the variability of outcomes of surgeries to restore thumb function. We lack a realistic biomechanical model of the thumb because of the variability/uncertainty of musculoskeletal parameters, the multiple proposed kinematic descriptions and methods to solve the muscle redundancy problem, and the paucity of data to validate the model with in vivo coordination patterns and force output. We performed a multi-stage validation of a biomechanical computer model against our measurements of maximal static thumbtip force and fine-wire electromyograms (EMG) from 8 thumb muscles in each of five orthogonal directions in key and opposition pinch postures. A low-friction point-contact at the thumbtip ensured that subjects did not produce thumbtip torques during force production. The 3-D, 8-muscle biomechanical thumb model uses a 5-axis kinematic description with orthogonal and intersecting axes of rotation at the carpometacarpal and metacarpophalangeal joints. We represented the 50 musculoskeletal parameters of the model as stochastic variables based on experimental data, and ran Monte Carlo simulations in the “inverse” and “forward” directions for 5000 random instantiations of the model. Two inverse simulations (predicting the distribution of maximal static thumbtip forces and the muscle activations that maximized force) showed that: the model reproduces at most 50% of the 80 EMG distributions recorded (eight muscle excitations in 5 force directions in two postures); and well-directed thumbtip forces of adequate magnitude are predicted only if accompanied by unrealistically large thumbtip torques (0.64±0.28 N m). The forward simulation (which fed the experimental distributions of EMG through random instantiations of the model) resulted in misdirected thumbtip force vectors (within 74.3±24.5° from the desired direction) accompanied by doubly large thumbtip torques (1.32±0.95 N m). Taken together, our results suggest that the variability and uncertainty of musculoskeletal parameters and the choice of solution method are not the likely reason for the unrealistic predictions obtained. Rather, the kinematic description of the thumb we used is not representative of the transformation of net joint torques into thumbtip forces/torques in the human thumb. Future efforts should focus on validating alternative kinematic descriptions of the thumb.