The human footprint is having a unique set of ridges unmatched by any other human being, and therefore it can be used in different identity documents for example birth certificate, Indian biometric identification system AADHAR card, driving license, PAN card, and passport. There are many instances of the crime scene where an accused must walk around and left the footwear impressions as well as barefoot prints and therefore, it is very crucial to recovering the footprints from identifying the criminals. Footprint-based biometric is a considerably newer technique for personal identification. Fingerprints, retina, iris and face recognition are the methods most useful for attendance record of the person. This time the world is facing the problem of global terrorism. It is challenging to identify the terrorist because they are living as regular as the citizens do. Their soft target includes the industries of special interests such as defence, silicon and nanotechnology chip manufacturing units, pharmacy sectors. They pretend themselves as religious persons, so temples and other holy places, even in markets is in their targets. These are the places where one can obtain their footprints quickly. The gait itself is sufficient to predict the behaviour of the suspects. The present research is driven to identify the usefulness of footprint and gait as an alternative to personal identification.

We introduce an unsupervised clustering algorithm to improve training efficiency and accuracy in predicting energies using molecular-orbital-based machine learning (MOB-ML). This work determines clusters via the Gaussian mixture model (GMM) in an entirely automatic manner and simplifies an earlier supervised clustering approach [J. Chem. Theory Comput., 15, 6668 (2019)] by eliminating both the necessity for user-specified parameters and the training of an additional classifier. Unsupervised clustering results from GMM have the advantage of accurately reproducing chemically intuitive groupings of frontier molecular orbitals and having improved performance with an increasing number of training examples. The resulting clusters from supervised or unsupervised clustering is further combined with scalable Gaussian process regression (GPR) or linear regression (LR) to learn molecular energies accurately by generating a local regression model in each cluster. Among all four combinations of regressors and clustering methods, GMM combined with scalable exact Gaussian process regression (GMM/GPR) is the most efficient training protocol for MOB-ML. The numerical tests of molecular energy learning on thermalized datasets of drug-like molecules demonstrate the improved accuracy, transferability, and learning efficiency of GMM/GPR over not only other training protocols for MOB-ML, i.e., supervised regression-clustering combined with GPR(RC/GPR) and GPR without clustering. GMM/GPR also provide the best molecular energy predictions compared with the ones from literature on the same benchmark datasets. With a lower scaling, GMM/GPR has a 10.4-fold speedup in wall-clock training time compared with scalable exact GPR with a training size of 6500 QM7b-T molecules.

Probability distributions play a fundamental role in statistical data analysis where, for continuous data, the dominant assumption is to consider random variables in Euclidean space. However, the Euclidean assumption is not appropriate for some data types and this motivates the development and study of distributions in non-Euclidean spaces. Notable examples include directional statistics (see [21]), in which observations typically lie on a sphere, and data that are expressed as tensors, such as covariance matrices ([32, 35, 36]) and data structures which arise in image and signal processing applications (see [2, 3]). Furthermore, in recent years, latent variable models have been shown to offer superior performance when the parameters are modelled in non-Euclidean spaces. Variational autoencoders (see [16, 27, 40, 23, 30]) and latent space network models (see [14, 19, 41, 24, 22]) offer two pertinent examples where it is most typical for non-Euclidean latent variables to be modelled in spherical or hyperbolic space.

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm that incorporates the gradient of the logarithm of the target density in its proposal distribution. In an earlier joint work \cite{pill:stu:12}, the author had extended the seminal work of \cite{Robe:Rose:98} and showed that in stationarity, MALA applied to an $N$-dimensional approximation of the target will take ${\cal O}(N^{\frac13})$ steps to explore its target measure. It was also shown in \cite{Robe:Rose:98,pill:stu:12} that, as a consequence of the diffusion limit, the MALA algorithm is optimized at an average acceptance probability of $0.574$. In \cite{pere:16}, Pereyra introduced the proximal MALA algorithm where the gradient of the log target density is replaced by the proximal function (mainly aimed at implementing MALA non-differentiable target densities). In this paper, we show that for a wide class of twice differentiable target densities, the proximal MALA enjoys the same optimal scaling as that of MALA in high dimensions and also has an average optimal acceptance probability of $0.574$. The results of this paper thus give the following practically useful guideline: for smooth target densities where it is expensive to compute the gradient while implementing MALA, users may replace the gradient with the corresponding proximal function (that can be often computed relatively cheaply via convex optimization) \emph{without} losing any efficiency. This confirms some of the empirical observations made in \cite{pere:16}.

The practical importance of coherent forecasts in hierarchical forecasting has inspired many studies on forecast reconciliation. Under this approach, so-called base forecasts are produced for every series in the hierarchy and are subsequently adjusted to be coherent in a second reconciliation step. Reconciliation methods have been shown to improve forecast accuracy, but will, in general, adjust the base forecast of every series. However, in an operational context, it is sometimes necessary or beneficial to keep forecasts of some variables unchanged after forecast reconciliation. In this paper, we formulate reconciliation methodology that keeps forecasts of a pre-specified subset of variables unchanged or "immutable". In contrast to existing approaches, these immutable forecasts need not all come from the same level of a hierarchy, and our method can also be applied to grouped hierarchies. We prove that our approach preserves unbiasedness in base forecasts. Our method can also account for correlations between base forecasting errors and ensure non-negativity of forecasts. We also perform empirical experiments, including an application to sales of a large scale online retailer, to assess the impacts of our proposed methodology.

Machine learning has been increasingly applied to solve forecasting problems. Classical forecasting approaches, such as ARIMA or exponential smoothing are being replaced by machine learning regression algorithms, such as XGBoost, Gaussian processes or deep learning. However, despite the increasing attention, there are still doubts about the forecasting performance of machine learning methods. Makridakis, one of the most prominent names in the forecasting literature, has recently presented evidence that classical methods systematically outperform machine learning approaches for univariate time series forecasting [1]. This includes algorithms such as the LSTM, multi-layer perceptron or Gaussian processes.

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In this work we describe an Adaptive Regularization using Cubics (ARC) method for large-scale nonconvex unconstrained optimization using Limited-memory Quasi-Newton (LQN) matrices. ARC methods are a relatively new family of optimization strategies that utilize a cubic-regularization (CR) term in place of trust-regions and line-searches. LQN methods offer a large-scale alternative to using explicit second-order information by taking identical inputs to those used by popular first-order methods such as stochastic gradient descent (SGD). Solving the CR subproblem exactly requires Newton's method, yet using properties of the internal structure of LQN matrices, we are able to find exact solutions to the CR subproblem in a matrix-free manner, providing large speedups and scaling into modern size requirements. Additionally, we expand upon previous ARC work and explicitly incorporate first-order updates into our algorithm. We provide experimental results when the SR1 update is used, which show substantial speed-ups and competitive performance compared to Adam and other second order optimizers on deep neural networks (DNNs). We find that our new approach, ARCLQN, compares to modern optimizers with minimal tuning, a common pain-point for second order methods.

Data that exhibits class imbalance appears frequently in real-world scenarios [1], in varying domains and applications: detecting pathologies or diseases in medical records [2], preventing network attacks in cybersecurity [3], detecting fraudulent financial transactions [4], distinguishing between earthquakes and explosions [5] and detecting spam communications [6]. In addition to these applications, where the class distribution is naturally skewed due to the frequency of events, some applications may exhibit class imbalance caused by extrinsic factors such as collection and storage limitations [7]. Most standard classification models are designed around and implicitly assume a relatively balanced class distribution; when applied without proper adjustments they may fail to accurately model the minority class and converge on a solution that over-classifies the majority class due to its increased prior probability [8]. These models thus neglect recall on the minority class and lead to unsatisfactory results when we desire high performance on a more balanced testing criterion. This issue is exacerbated by the fact that commonly used metrics such as accuracy may be misleading in evaluating the performance of the model. Even models that naively classify all samples as majority may have high accuracy under severe class imbalance. Most approaches to dealing with these shortcomings fall broadly into two categories: re-weighting the loss objective to more heavily account for the minority class, and resampling the input dataset such that the minority class is more prominent.

Stein variational gradient descent (SVGD) is a general-purpose optimization-based sampling algorithm that has recently exploded in popularity, but is limited by two issues: it is known to produce biased samples, and it can be slow to converge on complicated distributions. A recently proposed stochastic variant of SVGD (sSVGD) addresses the first issue, producing unbiased samples by incorporating a special noise into the SVGD dynamics such that asymptotic convergence is guaranteed. Meanwhile, Stein variational Newton (SVN), a Newton-like extension of SVGD, dramatically accelerates the convergence of SVGD by incorporating Hessian information into the dynamics, but also produces biased samples. In this paper we derive, and provide a practical implementation of, a stochastic variant of SVN (sSVN) which is both asymptotically correct and converges rapidly. We demonstrate the effectiveness of our algorithm on a difficult class of test problems -- the Hybrid Rosenbrock density -- and show that sSVN converges using three orders of magnitude fewer gradient evaluations of the log likelihood than its stochastic SVGD counterpart. Our results show that sSVN is a promising approach to accelerating high-precision Bayesian inference tasks with modest-dimension, $d\sim\mathcal{O}(10)$.