When it comes to machine learning algorithms, kernels play a crucial role in performing various tasks, such as classification, regression, and clustering. A kernel is a similarity function that takes two inputs and calculates the similarity between them. There are different types of kernels available in machine learning, but the Radial Basis Function (RBF) kernel is one of the most popular kernels. In this article, we will discuss the RBF kernel in detail and its importance in machine learning. The kernel trick is a powerful technique in machine learning that allows us to map data to higher dimensions without actually computing them.

In Part 1 we'll introduce the machine learning workflow and common techniques for cleaning and preparing raw data for analysis. We'll explore univariate analysis with frequency tables, histograms, kernel densities, and profiling metrics, then dive into multivariate profiling tools like heat maps, violin and box plots, scatter plots, and correlation: Variable types, empty values, range and count calculations, left/right censoring, etc. Histograms, frequency tables, mean, median, mode, variance, skewness, etc. Throughout the course, we'll introduce real-world scenarios to solidify key concepts and simulate actual data science and business intelligence cases. You'll use profiling metrics to clean up product inventory data for a local grocery, explore Olympic athlete demographics with histograms and kernel densities, visualize traffic accident frequency with heat maps, and more. In Part 2 we'll introduce the supervised learning landscape, review the classification workflow, and address key topics like dependent vs. independent variables, feature engineering, data splitting and overfitting.

Bayesian Optimization (BO) (Shahriari et al., 2015) is a popular sample-efficient optimization technique to solve problems where the objective is expensive. It has been successfully applied in diverse areas (Greenhill et al., 2020) including material discovery (Li et al., 2017), alloy design (Barnett et al., 2020) and molecular design (Gómez-Bombarelli et al., 2018). However, standard BO typically operates tabula rasa, building its model of the objective from minimal priors that do not include domain-specific information. While there has been some progress made incorporating domain-specific knowledge to accelerate BO (Li et al., 2018; Hvarfner et al., 2022) or transfer learning from previous experiments (Shilton et al., 2017), it remains the case that there is a significant corpus of knowledge and expertise that could potentially accelerate BO even further but which remain largely untapped due to the inherent complexities involved in knowledge extraction and exploitation. In particular, this often arises from the fact that experts tend to organize their knowledge in complex schema containing concepts, attributes and relationships (Rousseau, 2001), making the elicitation of relevant expert knowledge, both quantitative and qualitative, a difficult task.

In many applications, multiple parties have private data regarding the same set of users but on disjoint sets of attributes, and a server wants to leverage the data to train a model. To enable model learning while protecting the privacy of the data subjects, we need vertical federated learning (VFL) techniques, where the data parties share only information for training the model, instead of the private data. However, it is challenging to ensure that the shared information maintains privacy while learning accurate models. To the best of our knowledge, the algorithm proposed in this paper is the first practical solution for differentially private vertical federated k-means clustering, where the server can obtain a set of global centers with a provable differential privacy guarantee. Our algorithm assumes an untrusted central server that aggregates differentially private local centers and membership encodings from local data parties. It builds a weighted grid as the synopsis of the global dataset based on the received information. Final centers are generated by running any k-means algorithm on the weighted grid. Our approach for grid weight estimation uses a novel, light-weight, and differentially private set intersection cardinality estimation algorithm based on the Flajolet-Martin sketch. To improve the estimation accuracy in the setting with more than two data parties, we further propose a refined version of the weights estimation algorithm and a parameter tuning strategy to reduce the final k-means utility to be close to that in the central private setting. We provide theoretical utility analysis and experimental evaluation results for the cluster centers computed by our algorithm and show that our approach performs better both theoretically and empirically than the two baselines based on existing techniques.

In this work, we develop methods for few-shot image classification from a new perspective of optimal matching between image regions. We employ the Earth Mover's Distance (EMD) as a metric to compute a structural distance between dense image representations to determine image relevance. The EMD generates the optimal matching flows between structural elements that have the minimum matching cost, which is used to calculate the image distance for classification. To generate the important weights of elements in the EMD formulation, we design a cross-reference mechanism, which can effectively alleviate the adverse impact caused by the cluttered background and large intra-class appearance variations. To implement k-shot classification, we propose to learn a structured fully connected layer that can directly classify dense image representations with the EMD. Based on the implicit function theorem, the EMD can be inserted as a layer into the network for end-to-end training. Our extensive experiments validate the effectiveness of our algorithm which outperforms state-of-the-art methods by a significant margin on five widely used few-shot classification benchmarks, namely, miniImageNet, tieredImageNet, Fewshot-CIFAR100 (FC100), Caltech-UCSD Birds-200-2011 (CUB), and CIFAR-FewShot (CIFAR-FS). We also demonstrate the effectiveness of our method on the image retrieval task in our experiments.

We study stochastic zeroth order gradient and Hessian estimators for real-valued functions in $\mathbb{R}^n$. We show that, via taking finite difference along random orthogonal directions, the variance of the stochastic finite difference estimators can be significantly reduced. In particular, we design estimators for smooth functions such that, if one uses $ \Theta \left( k \right) $ random directions sampled from the Stiefel's manifold $ \text{St} (n,k) $ and finite-difference granularity $\delta$, the variance of the gradient estimator is bounded by $ \mathcal{O} \left( \left( \frac{n}{k} - 1 \right) + \left( \frac{n^2}{k} - n \right) \delta^2 + \frac{ n^2 \delta^4 }{ k } \right) $, and the variance of the Hessian estimator is bounded by $\mathcal{O} \left( \left( \frac{n^2}{k^2} - 1 \right) + \left( \frac{n^4}{k^2} - n^2 \right) \delta^2 + \frac{n^4 \delta^4 }{k^2} \right) $. When $k = n$, the variances become negligibly small. In addition, we provide improved bias bounds for the estimators. The bias of both gradient and Hessian estimators for smooth function $f$ is of order $\mathcal{O} \left( \delta^2 \Gamma \right)$, where $\delta$ is the finite-difference granularity, and $ \Gamma $ depends on high order derivatives of $f$. Our results are evidenced by empirical observations.

Objective. Research on brain-computer interfaces (BCIs) is advancing towards rehabilitating severely disabled patients in the real world. Two key factors for successful decoding of user intentions are the size of implanted microelectrode arrays and a good online spike sorting algorithm. A small but dense microelectrode array with 3072 channels was recently developed for decoding user intentions. The process of spike sorting determines the spike activity (SA) of different sources (neurons) from recorded neural data. Unfortunately, current spike sorting algorithms are unable to handle the massively increasing amount of data from dense microelectrode arrays, making spike sorting a fragile component of the online BCI decoding framework. Approach. We proposed an adaptive and self-organized algorithm for online spike sorting, named Adaptive SpikeDeep-Classifier (Ada-SpikeDeepClassifier), which uses SpikeDeeptector for channel selection, an adaptive background activity rejector (Ada-BAR) for discarding background events, and an adaptive spike classifier (Ada-Spike classifier) for classifying the SA of different neural units. Results. Our algorithm outperformed our previously published SpikeDeep-Classifier and eight other spike sorting algorithms, as evaluated on a human dataset and a publicly available simulated dataset. Significance. The proposed algorithm is the first spike sorting algorithm that automatically learns the abrupt changes in the distribution of noise and SA. It is an artificial neural network-based algorithm that is well-suited for hardware implementation on neuromorphic chips that can be used for wearable invasive BCIs.

Multi-label learning is usually used to mine the correlation between features and labels, and feature selection can retain as much information as possible through a small number of features. $\ell_{2,1}$ regularization method can get sparse coefficient matrix, but it can not solve multicollinearity problem effectively. The model proposed in this paper can obtain the most relevant few features by solving the joint constrained optimization problems of $\ell_{2,1}$ and $\ell_{F}$ regularization.In manifold regularization, we implement random walk strategy based on joint information matrix, and get a highly robust neighborhood graph.In addition, we given the algorithm for solving the model and proved its convergence.Comparative experiments on real-world data sets show that the proposed method outperforms other methods.

The success of existing multi-view clustering relies on the assumption of sample integrity across multiple views. However, in real-world scenarios, samples of multi-view are partially available due to data corruption or sensor failure, which leads to incomplete multi-view clustering study (IMVC). Although several attempts have been proposed to address IMVC, they suffer from the following drawbacks: i) Existing methods mainly adopt cross-view contrastive learning forcing the representations of each sample across views to be exactly the same, which might ignore view discrepancy and flexibility in representations; ii) Due to the absence of non-observed samples across multiple views, the obtained prototypes of clusters might be unaligned and biased, leading to incorrect fusion. To address the above issues, we propose a Cross-view Partial Sample and Prototype Alignment Network (CPSPAN) for Deep Incomplete Multi-view Clustering. Firstly, unlike existing contrastive-based methods, we adopt pair-observed data alignment as 'proxy supervised signals' to guide instance-to-instance correspondence construction among views. Then, regarding of the shifted prototypes in IMVC, we further propose a prototype alignment module to achieve incomplete distribution calibration across views. Extensive experimental results showcase the effectiveness of our proposed modules, attaining noteworthy performance improvements when compared to existing IMVC competitors on benchmark datasets.

We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite B\' ezier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.