Model selection is crucial to high-dimensional learning and inference for contemporary big data applications in pinpointing the best set of covariates among a sequence of candidate interpretable models. Most existing work assumes implicitly that the models are correctly specified or have fixed dimensionality. Yet both features of model misspecification and high dimensionality are prevalent in practice. In this paper, we exploit the framework of model selection principles in misspecified models originated in Lv and Liu (2014) and investigate the asymptotic expansion of Bayesian principle of model selection in the setting of high-dimensional misspecified models. With a natural choice of prior probabilities that encourages interpretability and incorporates Kullback-Leibler divergence, we suggest the high-dimensional generalized Bayesian information criterion with prior probability (HGBIC_p) for large-scale model selection with misspecification. Our new information criterion characterizes the impacts of both model misspecification and high dimensionality on model selection. We further establish the consistency of covariance contrast matrix estimation and the model selection consistency of HGBIC_p in ultra-high dimensions under some mild regularity conditions. The advantages of our new method are supported by numerical studies.

This paper considers the problem of estimating a high-dimensional vector of parameters $\boldsymbol{\theta} \in \mathbb{R}^n$ from a noisy observation. The noise vector is i.i.d. Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension $n$ exceeds two. The JS-estimator shrinks the observed vector towards the origin, and the risk reduction over the ML-estimator is greatest for $\boldsymbol{\theta}$ that lie close to the origin. JS-estimators can be generalized to shrink the data towards any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when $\boldsymbol{\theta}$ lies close to the subspace. This leads to the question: in the absence of prior information about $\boldsymbol{\theta}$, how do we design estimators that give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$? In this paper, we propose shrinkage estimators that attempt to infer the structure of $\boldsymbol{\theta}$ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk towards a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$, particularly for large $n$. Simulation results are provided to support the theoretical claims.

Statistical cluster analysis is a Exploratory Data Analysis Technique which groups heterogeneous objects (M.D.) into homogeneous groups. We will learn the basics of cluster analysis with mathematical way. Note: Result of both the approaches are displayed through the dendrogram tree. Hierarchical cluster analysis ends here, in the next tutorial article I will explain Non-Hierarchical cluster analysis. Till then stay tuned and keep visiting for learning tutorials which you won't get anywhere.

At the end of my previous post on Which Continent Does PyoungChang Belong To? I made a simple-voting-based ensemble function. Given city name, the ensemble model loads five models and aggregates their inference results, and chooses the most voted continent as its final prediction. This ensemble model's f1-score was 0.571 which was 0.005 higher than the single model's 0.566. Taking advantage of the collaborative power is simple to build and easy to use, but it forced me to set up and train multiple models.

Anaconda is used for managing my dependencies. You can download the weight from here and save it to the weights folder. The code for this step is contained in the function named extract_features and codes from line 464 to 552 in svm_pipeline.py. If the SVM classifier exist, load it directly. Otherwise, I started by reading in all the vehicle and non-vehicle images, around 8000 images in each category.

Let's say I am given an Excel sheet with data about various fruits and I have to tell which look like Apples. What I will do is ask a question "Which fruits are red and round?" and divide all fruits which answer yes and no to the question. Now, All Red and Round fruits might not be apples and all apples won't be red and round. So I will ask a question "Which fruits have red or yellow color hints on them? " on red and round fruits and will ask "Which fruits are green and round?" on not red and round fruits. Based on these questions I can tell with considerable accuracy which are apples. This cascade of questions is what a decision tree is. However, this is a decision tree based on my intuition.

Originally introduced by [7], Random Forests (RF) are state-of-the-art classification and regression algorithms that proceed by averaging the forecasts of a number of randomized decision trees grown in parallel. Despite their widespread use and remarkable success in practical applications, the theoretical properties of such algorithms are still not fully understood. For an overview of theoretical results on random forests, see [5]. As a result of the complexity of the procedure, which combines sampling steps and feature selection, Breiman's original algorithm has proved difficult to analyze. Consequently, most theoretical studies focus on modified and stylized versions of Random Forests. Among these methods, Purely Random Forests (PRF) [6, 4, 3, 13, 2] that grow the individual trees independently of the sample, are particularly amenable to theoretical analysis. The consistency of such estimates (as well as other idealized RF procedures) was first obtained by [4], as a byproduct of the consistency of individual tree estimates. A recent line of research [25, 28, 18, 27] has sought to obtain some theoretical guarantees for RF variants that more closely resembled the algorithm used in practice. It should be noted, however, that most of these theoretical guarantees come at the price of assumptions either on the data structure or on the Random Forest algorithm itself, being thus still far from explaining the excellent empirical performance of Random Forests.

We consider the problem of bipartite community detection in networks, or more generally the network biclustering problem. We present a fast two-stage procedure based on spectral initialization followed by the application of a pseudo-likelihood classifier twice. Under mild regularity conditions, we establish the weak consistency of the procedure (i.e., the convergence of the misclassification rate to zero) under a general bipartite stochastic block model. We show that the procedure is optimal in the sense that it achieves the optimal convergence rate that is achievable by a biclustering oracle, adaptively over the whole class, up to constants. The optimal rate we obtain sharpens some of the existing results and generalizes others to a wide regime of average degree growth. As a special case, we recover the known exact recovery threshold in the $\log n$ regime of sparsity. To obtain the general consistency result, as part of the provable version of the algorithm, we introduce a sub-block partitioning scheme that is also computationally attractive, allowing for distributed implementation of the algorithm without sacrificing optimality. The provable version of the algorithm is derived from a general blueprint for pseudo-likelihood biclustering algorithms that employ simple EM type updates. We show the effectiveness of this general class by numerical simulations.

Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results. We provide provable guarantees for deterministic column sampling using ridge leverage scores. The matrix sketch returned by our algorithm is a column subset of the original matrix, yielding additional interpretability. Like the randomized counterparts, the deterministic algorithm provides (1 + {\epsilon}) error column subset selection, (1 + {\epsilon}) error projection-cost preservation, and an additive-multiplicative spectral bound. We also show that under the assumption of power-law decay of ridge leverage scores, this deterministic algorithm is provably as accurate as randomized algorithms. Lastly, ridge regression is frequently used to regularize ill-posed linear least- squares problems. While ridge regression provides shrinkage for the regression coefficients, many of the coefficients remain small but non-zero. Performing ridge regression with the matrix sketch returned by our algorithm and a particular regularization parameter forces coefficients to zero and has a provable (1 + {\epsilon}) bound on the statistical risk. As such, it is an interesting alternative to elastic net regularization.

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this observation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.