The backtracking line-search is an effective technique to automatically tune the step-size in smooth optimization. It guarantees similar performance to using the theoretically optimal step-size. Many approaches have been developed to instead tune per-coordinate step-sizes, also known as diagonal preconditioners, but none of the existing methods are provably competitive with the optimal per-coordinate stepsizes. We propose multidimensional backtracking, an extension of the backtracking line-search to find good diagonal preconditioners for smooth convex problems. Our key insight is that the gradient with respect to the step-sizes, also known as hypergradients, yields separating hyperplanes that let us search for good preconditioners using cutting-plane methods. As black-box cutting-plane approaches like the ellipsoid method are computationally prohibitive, we develop an efficient algorithm tailored to our setting. Multidimensional backtracking is provably competitive with the best diagonal preconditioner and requires no manual tuning.

We introduce Non-Euc lidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

Ensuring liveability and comfort is one of the fundamental objectives of urban planning. Numerous studies have employed computational methods to assess and quantify factors related to urban comfort such as greenery coverage, thermal comfort, and walkability. However, a clear definition of urban comfort and its comprehensive evaluation framework remain elusive. Our research explores the theoretical interpretations and methodologies for assessing urban comfort within digital planning, emphasising three key dimensions: multidimensional analysis, data support, and AI assistance.

The classical multidimensional scaling (cMDS) algorithm is a widespread method to do this.

The work extends a recent publication on the same topic in three ways: (1) it incorporates a low-rank regularisation on the network output, (2) adds another set of latent variables to match the distribution of Hebbian/Anti-Hebbian plasticity rules in cortical networks and (3) adds a penalty term to whiten the outputs.

We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

Intransitivity is a critical issue in pairwise preference modeling. It refers to the intransitive pairwise preferences between a group of players or objects that potentially form a cyclic preference chain and has been long discussed in social choice theory in the context of the dominance relationship. However, such multifaceted intransitivity between players and the corresponding player representations in high dimensions is difficult to capture. In this paper, we propose a probabilistic model that jointly learns each player's d-dimensional representation (d>1) and a dataset-specific metric space that systematically captures the distance metric in Rd over the embedding space. Interestingly, by imposing additional constraints in the metric space, our proposed model degenerates to former models used in intransitive representation learning. Moreover, we present an extensive quantitative investigation of the vast existence of intransitive relationships between objects in various real-world benchmark datasets. To our knowledge, this investigation is the first of this type. The predictive performance of our proposed method on different real-world datasets, including social choice, election, and online game datasets, shows that our proposed method outperforms several competing methods in terms of prediction accuracy.

Educational Data Mining (EDM) is a promising field, where data mining is widely used for predicting students performance. One of the most prevalent and recent challenge that higher education faces today is making students skillfully employable. Institutions possess large volume of data; still they are unable to reveal knowledge and guide their students. Data in education is generally very large, multidimensional and unbalanced in nature. Process of extracting knowledge from such data has its own set of problems and is a very complicated task. In this paper, Engineering and MCA (Masters in Computer Applications) students data is collected from various universities and institutes pan India. The dataset is large, unbalanced and multidimensional in nature. A cluster based model is presented in this paper, which, when applied at preprocessing stage helps in parsimonious selection of variables and improves the performance of predictive algorithms. Hence, facilitate in better prediction of Students Employability.