Computational geometry

We propose an independence criterion based on the eigen-spectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. First, the empirical estimate is simpler than any other kernel dependence test, and requires no user-defined regularisation. Second, there is a clearly defined population quantity which the empirical estimate approaches in the large sample limit, with exponential convergence guaranteed between the two: this ensures that independence tests based on HSIC do not suffer from slow learning rates. Finally, we show in the context of independent component analysis (ICA) that the performance of HSIC is competitive with that of previously published kernel-based criteria, and of other recently published ICA methods.

Measuring statistical dependence with Hilbert-Schmidt norms

Pattern recognition

Learning and then recognizing a route, whether travelled during the day or at night, in clear or inclement weather, and in summer or winter is a challenging task for state of the art algorithms in computer vision and robotics. In this paper, we present a new approach to visual navigation under changing conditions dubbed SeqSLAM. Instead of calculating the single location most likely given a current image, our approach calculates the best candidate matching location within every local navigation sequence. Localization is then achieved by recognizing coherent sequences of these “local best matches”. This approach removes the need for global matching performance by the vision front-end - instead it must only pick the best match within any short sequence of images. The approach is applicable over environment changes that render traditional feature-based techniques ineffective. Using two car-mounted camera datasets we demonstrate the effectiveness of the algorithm and compare it to one of the most successful feature-based SLAM algorithms, FAB-MAP. The perceptual change in the datasets is extreme; repeated traverses through environments during the day and then in the middle of the night, at times separated by months or years and in opposite seasons, and in clear weather and extremely heavy rain. While the feature-based method fails, the sequence-based algorithm is able to match trajectory segments at 100% precision with recall rates of up to 60%.

SeqSLAM : visual route-based navigation for sunny summer days and stormy winter nights

College Physics Select one of the following Physics sequences: 10 Algebraand Trigonometry-based: PHSX 205N & PHSX 206N College Physics I and College Physics I Laboratory PHSX 207N & PHSX 208N College Physics II and College Physics II Laboratory Calculus-based (strongly recommended): PHSX 215N & PHSX 216N Fund of Physics w/Calc I and Physics Laboratory I w/Calc PHSX 217N & PHSX 218N Fund of Physics w/Calc II and Physics Laboratory II w/Calc Total Hours 10

COMPUTATIONAL Physics

In this tutorial paper, we first define mean squared error, variance, covariance, and bias of both random variables and classification/predictor models. Then, we formulate the true and generalization errors of the model for both training and validation/test instances where we make use of the Stein's Unbiased Risk Estimator (SURE). We define overfitting, underfitting, and generalization using the obtained true and generalization errors. We introduce cross validation and two well-known examples which are $K$-fold and leave-one-out cross validations. We briefly introduce generalized cross validation and then move on to regularization where we use the SURE again. We work on both $\ell_2$ and $\ell_1$ norm regularizations. Then, we show that bootstrap aggregating (bagging) reduces the variance of estimation. Boosting, specifically AdaBoost, is introduced and it is explained as both an additive model and a maximum margin model, i.e., Support Vector Machine (SVM). The upper bound on the generalization error of boosting is also provided to show why boosting prevents from overfitting. As examples of regularization, the theory of ridge and lasso regressions, weight decay, noise injection to input/weights, and early stopping are explained. Random forest, dropout, histogram of oriented gradients, and single shot multi-box detector are explained as examples of bagging in machine learning and computer vision. Finally, boosting tree and SVM models are mentioned as examples of boosting.

The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also provide examples from machine learning, including principal component analysis, kernel supervised principal component analysis, and Fisher discriminant analysis, which result in eigenvalue and generalized eigenvalue problems. Finally, we introduce the solutions to both eigenvalue and generalized eigenvalue problems.

Eigenvalue and Generalized Eigenvalue Problems: Tutorial

Pattern analysis often requires a pre-processing stage for extracting or selecting features in order to help the classification, prediction, or clustering stage discriminate or represent the data in a better way. The reason for this requirement is that the raw data are complex and difficult to process without extracting or selecting appropriate features beforehand. This paper reviews theory and motivation of different common methods of feature selection and extraction and introduces some of their applications. Some numerical implementations are also shown for these methods. Finally, the methods in feature selection and extraction are compared.

Feature Selection and Feature Extraction in Pattern Analysis: A Literature Review.

This tutorial explains Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) as two fundamental classification methods in statistical and probabilistic learning. We start with the optimization of decision boundary on which the posteriors are equal. Then, LDA and QDA are derived for binary and multiple classes. The estimation of parameters in LDA and QDA are also covered. Then, we explain how LDA and QDA are related to metric learning, kernel principal component analysis, Mahalanobis distance, logistic regression, Bayes optimal classifier, Gaussian naive Bayes, and likelihood ratio test. We also prove that LDA and Fisher discriminant analysis are equivalent. We finally clarify some of the theoretical concepts with simulations we provide.

Linear and Quadratic Discriminant Analysis: Tutorial.

This paper proposes a new method for view-invariant action recognition that utilizes the temporal position of skeletal joints obtained by Kinect sensor. In this method, the actions are represented as sequences of several pre-defined poses. After pre-processing, which includes skeleton alignment and scaling, the appropriate feature vectors are obtained for recognizing and discriminating the pose of every frame by the proposed Fisherposes method. The proposed regularized Mahalanobis distance metric is used in order to recognize both the involuntary and highly made-up actions at the same time. Hidden Markov model (HMM) is then used to classify the action related to an input sequence of poses. For taking into account the motion in the actions which are not separable by solely their temporal poses, histograms of trajectories are also proposed. The proposed action recognition method is capable of recognizing both the voluntary and involuntary actions, as well as pose-based and trajectory-based ones with a high accuracy rate. The effectiveness of the proposed method is experimented on three publicly available data sets, TST fall detection, UTKinect, and UCFKinect data sets.

Benyamin Ghojogh

Papers

The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial

Eigenvalue and Generalized Eigenvalue Problems: Tutorial

Feature Selection and Feature Extraction in Pattern Analysis: A Literature Review.

Linear and Quadratic Discriminant Analysis: Tutorial.

Fisherposes for Human Action Recognition Using Kinect Sensor Data