We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.

We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for this behavior. Motivated by this discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation. The method reduces the time cost from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \log d)$, where $d$ is the data dimensionality, with almost no compromise in kernel approximation quality compared to ORF. Experiments on several datasets verify the effectiveness of ORF and SORF over the existing methods. We also provide discussions on using the same type of discrete orthogonal structure for a broader range of applications.

We present a generic compact computational framework relying on structured random matrices that can be applied to speed up several machine learning algorithms with almost no loss of accuracy. The applications include new fast LSH-based algorithms, efficient kernel computations via random feature maps, convex optimization algorithms, quantization techniques and many more. Certain models of the presented paradigm are even more compressible since they apply only bit matrices. This makes them suitable for deploying on mobile devices. All our findings come with strong theoretical guarantees. In particular, as a byproduct of the presented techniques and by using relatively new Berry-Esseen-type CLT for random vectors, we give the first theoretical guarantees for one of the most efficient existing LSH algorithms based on the $\textbf{HD}_{3}\textbf{HD}_{2}\textbf{HD}_{1}$ structured matrix ("Practical and Optimal LSH for Angular Distance"). These guarantees as well as theoretical results for other aforementioned applications follow from the same general theoretical principle that we present in the paper. Our structured family contains as special cases all previously considered structured schemes, including the recently introduced $P$-model. Experimental evaluation confirms the accuracy and efficiency of TripleSpin matrices.