Density peaks clustering has become a nova of clustering algorithm because of its simplicity and practicality. However, there is one main drawback: it is time-consuming due to its high computational complexity. Herein, a density peaks clustering algorithm with sparse search and K-d tree is developed to solve this problem. Firstly, a sparse distance matrix is calculated by using K-d tree to replace the original full rank distance matrix, so as to accelerate the calculation of local density. Secondly, a sparse search strategy is proposed to accelerate the computation of relative-separation with the intersection between the set of $k$ nearest neighbors and the set consisting of the data points with larger local density for any data point. Furthermore, a second-order difference method for decision values is adopted to determine the cluster centers adaptively. Finally, experiments are carried out on datasets with different distribution characteristics, by comparing with other six state-of-the-art clustering algorithms. It is proved that the algorithm can effectively reduce the computational complexity of the original DPC from $O(n^2K)$ to $O(n(n^{1-1/K}+k))$. Especially for larger datasets, the efficiency is elevated more remarkably. Moreover, the clustering accuracy is also improved to a certain extent. Therefore, it can be concluded that the overall performance of the newly proposed algorithm is excellent.

Deep operator learning has emerged as a promising tool for reduced-order modelling and PDE model discovery. Leveraging the expressive power of deep neural networks, especially in high dimensions, such methods learn the mapping between functional state variables. While proposed methods have assumed noise only in the dependent variables, experimental and numerical data for operator learning typically exhibit noise in the independent variables as well, since both variables represent signals that are subject to measurement error. In regression on scalar data, failure to account for noisy independent variables can lead to biased parameter estimates. With noisy independent variables, linear models fitted via ordinary least squares (OLS) will show attenuation bias, wherein the slope will be underestimated. In this work, we derive an analogue of attenuation bias for linear operator regression with white noise in both the independent and dependent variables. In the nonlinear setting, we computationally demonstrate underprediction of the action of the Burgers operator in the presence of noise in the independent variable. We propose error-in-variables (EiV) models for two operator regression methods, MOR-Physics and DeepONet, and demonstrate that these new models reduce bias in the presence of noisy independent variables for a variety of operator learning problems. Considering the Burgers operator in 1D and 2D, we demonstrate that EiV operator learning robustly recovers operators in high-noise regimes that defeat OLS operator learning. We also introduce an EiV model for time-evolving PDE discovery and show that OLS and EiV perform similarly in learning the Kuramoto-Sivashinsky evolution operator from corrupted data, suggesting that the effect of bias in OLS operator learning depends on the regularity of the target operator.

Techniques of hybridisation and ensemble learning are popular model fusion techniques for improving the predictive power of forecasting methods. With limited research that instigates combining these two promising approaches, this paper focuses on the utility of the Exponential-Smoothing-Recurrent Neural Network (ES-RNN) in the pool of base models for different ensembles. We compare against some state of the art ensembling techniques and arithmetic model averaging as a benchmark. We experiment with the M4 forecasting data set of 100,000 time-series, and the results show that the Feature-based Forecast Model Averaging (FFORMA), on average, is the best technique for late data fusion with the ES-RNN. However, considering the M4's Daily subset of data, stacking was the only successful ensemble at dealing with the case where all base model performances are similar. Our experimental results indicate that we attain state of the art forecasting results compared to N-BEATS as a benchmark. We conclude that model averaging is a more robust ensemble than model selection and stacking strategies. Further, the results show that gradient boosting is superior for implementing ensemble learning strategies.

In the third of our article series, we touched on the topic of Linear Regression. We then used this method to create a house price estimator. The concepts of Linear Regression and Logistic Regression should not be confused with each other. Logistic Regression is a Linear Classifier. A statistical method used to analyze a dataset with one or more independent variables that determine a class. The outcome is measured with a binary variable (there are only two possible outcomes).

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Clustering is an unsupervised Machine Learning technique, where there is neither a training set nor predefined classes. Clustering is used when there are many records, which should be grouped according to similarity criteria, such as distance. A clustering algorithm takes a dataset as input and returns a list of labels as output, corresponding to the associated clusters. Cluster analysis is an iterative process where, at each step, the current iteration is evaluated and used to feedback into changes to the algorithm in the next iteration, until the desired result is obtained. The scikit-learn library provides a subpackage, called sklearn.cluster, which provides the most common clustering algorithms.

This article will explain some of the most well known machine learning algorithms in less than a minute - helping everyone to understand them! One of the simplest Machine learning algorithms out there, Linear Regression is used to make predictions on continuous dependent variables with knowledge from independent variables. A dependent variable is the effect, in which its value depends on changes in the independent variable. You may remember the line of best fit from school - this is what Linear Regression produces. A simple example is predicting one's weight depending on their height.

Macro placement is the problem of placing memory blocks on a chip canvas. It can be formulated as a combinatorial optimization problem over sequence pairs, a representation which describes the relative positions of macros. Solving this problem is particularly challenging since the objective function is expensive to evaluate. In this paper, we develop a novel approach to macro placement using Bayesian optimization (BO) over sequence pairs. BO is a machine learning technique that uses a probabilistic surrogate model and an acquisition function that balances exploration and exploitation to efficiently optimize a black-box objective function. BO is more sample-efficient than reinforcement learning and therefore can be used with more realistic objectives. Additionally, the ability to learn from data and adapt the algorithm to the objective function makes BO an appealing alternative to other black-box optimization methods such as simulated annealing, which relies on problem-dependent heuristics and parameter-tuning. We benchmark our algorithm on the fixed-outline macro placement problem with the half-perimeter wire length objective and demonstrate competitive performance.

This paper presents a multifunctional interdisciplinary framework that makes four scientific contributions towards the development of personalized ambient assisted living, with a specific focus to address the different and dynamic needs of the diverse aging population in the future of smart living environments. First, it presents a probabilistic reasoning-based mathematical approach to model all possible forms of user interactions for any activity arising from the user diversity of multiple users in such environments. Second, it presents a system that uses this approach with a machine learning method to model individual user profiles and user-specific user interactions for detecting the dynamic indoor location of each specific user. Third, to address the need to develop highly accurate indoor localization systems for increased trust, reliance, and seamless user acceptance, the framework introduces a novel methodology where two boosting approaches Gradient Boosting and the AdaBoost algorithm are integrated and used on a decision tree-based learning model to perform indoor localization. Fourth, the framework introduces two novel functionalities to provide semantic context to indoor localization in terms of detecting each user's floor-specific location as well as tracking whether a specific user was located inside or outside a given spatial region in a multi-floor-based indoor setting. These novel functionalities of the proposed framework were tested on a dataset of localization-related Big Data collected from 18 different users who navigated in 3 buildings consisting of 5 floors and 254 indoor spatial regions. The results show that this approach of indoor localization for personalized AAL that models each specific user always achieves higher accuracy as compared to the traditional approach of modeling an average user.

Admittedly, Graph Convolution Network (GCN) has achieved excellent results on graph datasets such as social networks, citation networks, etc. However, softmax used as the decision layer in these frameworks is generally optimized with thousands of iterations via gradient descent. Furthermore, due to ignoring the inner distribution of the graph nodes, the decision layer might lead to an unsatisfactory performance in semi-supervised learning with less label support. To address the referred issues, we propose a novel graph deep model with a non-gradient decision layer for graph mining. Firstly, manifold learning is unified with label local-structure preservation to capture the topological information of the nodes. Moreover, owing to the non-gradient property, closed-form solutions is achieved to be employed as the decision layer for GCN. Particularly, a joint optimization method is designed for this graph model, which extremely accelerates the convergence of the model. Finally, extensive experiments show that the proposed model has achieved state-of-the-art performance compared to the current models.