We focus on the problem of estimating the change in the dependency structures of two $p$-dimensional Gaussian Graphical models (GGMs). Previous studies for sparse change estimation in GGMs involve expensive and difficult non-smooth optimization. We propose a novel method, DIFFEE for estimating DIFFerential networks via an Elementary Estimator under a high-dimensional situation. DIFFEE is solved through a faster and closed form solution that enables it to work in large-scale settings. We conduct a rigorous statistical analysis showing that surprisingly DIFFEE achieves the same asymptotic convergence rates as the state-of-the-art estimators that are much more difficult to compute. Our experimental results on multiple synthetic datasets and one real-world data about brain connectivity show strong performance improvements over baselines, as well as significant computational benefits.

The goal in supervised learning is to learn from examples a function that predicts well new data. Nonparametric methods are often crucial since the functions to be learned can be nonlinear and complex Kernel methods are probably the most popular among nonparametric learning methods, but despite excellent theoretical properties, they have limited applications in large scale learning because of time and memory requirements, typically at least quadratic in the number of data points. Overcoming these limitations has motivated a variety of practical approaches including gradient methods, as well accelerated, stochastic and preconditioned extensions, to improve time complexity [1, 2, 3, 4, 5, 6]. Random projections provide an approach to reduce memory requirements, popular methods including Nyström [7, 8], random features [9], and their numerous extensions. From a theoretical perspective a key question has become to characterize statistical and computational tradeoffs, that is if, or under which conditions, computational gains come at the expense of statistical accuracy.

We study the generalization properties of ridge regression with random features in the statistical learning framework. We show for the first time that $O(1/\sqrt{n})$ learning bounds can be achieved with only $O(\sqrt{n}\log n)$ random features rather than $O({n})$ as suggested by previous results. Further, we prove faster learning rates and show that they might require more random features, unless they are sampled according to a possibly problem dependent distribution. Our results shed light on the statistical computational trade-offs in large scale kernelized learning, showing the potential effectiveness of random features in reducing the computational complexity while keeping optimal generalization properties.

Early stopping is a well known approach to reduce the time complexity for performing training and model selection of large scale learning machines. On the other hand, memory/space (rather than time) complexity is the main constraint in many applications, and randomized subsampling techniques have been proposed to tackle this issue. In this paper we ask whether early stopping and subsampling ideas can be combined in a fruitful way. We consider the question in a least squares regression setting and propose a form of randomized iterative regularization based on early stopping and subsampling. In this context, we analyze the statistical and computational properties of the proposed method. Theoretical results are complemented and validated by a thorough experimental analysis.

Electron Cryo-Tomography (ECT) allows 3D visualization of subcellular structures at the submolecular resolution in close to the native state. However, due to the high degree of structural complexity and imaging limits, the automatic segmentation of cellular components from ECT images is very difficult. To complement and speed up existing segmentation methods, it is desirable to develop a generic cell component segmentation method that is 1) not specific to particular types of cellular components, 2) able to segment unknown cellular components, 3) fully unsupervised and does not rely on the availability of training data. As an important step towards this goal, in this paper, we propose a saliency detection method that computes the likelihood that a subregion in a tomogram stands out from the background. Our method consists of four steps: supervoxel over-segmentation, feature extraction, feature matrix decomposition, and computation of saliency. The method produces a distribution map that represents the regions' saliency in tomograms. Our experiments show that our method can successfully label most salient regions detected by a human observer, and able to filter out regions not containing cellular components. Therefore, our method can remove the majority of the background region, and significantly speed up the subsequent processing of segmentation and recognition of cellular components captured by ECT.

From the above plot, it is evident that the clusters 1, 7, and 9 have activity for all 24 hours and are the more revenue generating clusters. The clusters 1, 5, 7, 9, and 10 have activity in night hours. The cluster 5 has activity from 11.5 to 17 hours.

In this previous post, I used linear regression to remove ocular artifacts from EEG signals. A popular alternative to this approach is independent component analysis (ICA). Components that represent ocular activity can be identified and eliminated to reconstruct artifact-free EEG signals. This approach is described in more detail in Jung et al. (2000). A comprehensive comparison between the two methods is beyond the scope of this post.

When we talk about Regression, we often end up discussing Linear and Logistics Regression. Do you know there are 7 types of Regressions? Linear and logistic regression is just the most loved members from the family of regressions. Last week, I saw a recorded talk at NYC Data Science Academy from Owen Zhang, current Kaggle rank 3 and Chief Product Officer at DataRobot. He said, 'if you are using regression without regularization, you have to be very special!'. I hope you get what a person of his stature referred to. I understood it very well and decided to explore regularization techniques in detail. In this article, I have explained the complex science behind'Ridge Regression' and'Lasso Regression' which are the most fundamental regularization techniques, sadly still not used by many.

Gaussian processes (GPs) are flexible models that can capture complex structure in large-scale dataset due to their non-parametric nature. However, the usage of GPs in real-world application is limited due to their high computational cost at inference time. In this paper, we introduce a new framework, \textit{kernel distillation}, for kernel matrix approximation. The idea adopts from knowledge distillation in deep learning community, where we approximate a fully trained teacher kernel matrix of size $n\times n$ with a student kernel matrix. We combine inducing points method with sparse low-rank approximation in the distillation procedure. The distilled student kernel matrix only cost $\mathcal{O}(m^2)$ storage where $m$ is the number of inducing points and $m \ll n$. We also show that one application of kernel distillation is for fast GP prediction, where we demonstrate empirically that our approximation provide better balance between the prediction time and the predictive performance compared to the alternatives.

We consider high dimensional dynamic multi-product pricing with an evolving but low-dimensional linear demand model. Assuming the temporal variation in cross-elasticities exhibits low-rank structure based on fixed (latent) features of the products, we show that the revenue maximization problem reduces to an online bandit convex optimization with side information given by the observed demands. We design dynamic pricing algorithms whose revenue approaches that of the best fixed price vector in hindsight, at a rate that only depends on the intrinsic rank of the demand model and not the number of products. Our approach applies a bandit convex optimization algorithm in a projected low-dimensional space spanned by the latent product features, while simultaneously learning this span via online singular value decomposition of a carefully-crafted matrix containing the observed demands.