Spectral Toolkit of Algorithms for Graphs (STAG) is an open-source C++ and Python library providing several methods for working with graphs and performing graph-based data analysis. In this technical report, we provide an update on the development of the STAG library. The report serves as a user's guide for the newly implemented algorithms, and gives implementation details and engineering choices made in the development of the library. The report is structured as follows: Section 2 describes the locality sensitive hashing, and the main components used in its construction. Section 3 describes the kernel density estimation, and the state-of-the-art algorithm for the kernel density estimation.

Feature preprocessing continues to play a critical role when applying machine learning and statistical methods to tabular data. In this paper, we propose the use of the kernel density integral transformation as a feature preprocessing step. Our approach subsumes the two leading feature preprocessing methods as limiting cases: linear min-max scaling and quantile transformation. We demonstrate that, without hyperparameter tuning, the kernel density integral transformation can be used as a simple drop-in replacement for either method, offering protection from the weaknesses of each. Alternatively, with tuning of a single continuous hyperparameter, we frequently outperform both of these methods. Finally, we show that the kernel density transformation can be profitably applied to statistical data analysis, particularly in correlation analysis and univariate clustering.

Dot-product attention mechanism plays a crucial role in modern deep architectures (e.g., Transformer) for sequence modeling, however, na\"ive exact computation of this model incurs quadratic time and memory complexities in sequence length, hindering the training of long-sequence models. Critical bottlenecks are due to the computation of partition functions in the denominator of softmax function as well as the multiplication of the softmax matrix with the matrix of values. Our key observation is that the former can be reduced to a variant of the kernel density estimation (KDE) problem, and an efficient KDE solver can be further utilized to accelerate the latter via subsampling-based fast matrix products. Our proposed KDEformer can approximate the attention in sub-quadratic time with provable spectral norm bounds, while all prior results merely provide entry-wise error bounds. Empirically, we verify that KDEformer outperforms other attention approximations in terms of accuracy, memory, and runtime on various pre-trained models. On BigGAN image generation, we achieve better generative scores than the exact computation with over $4\times$ speedup. For ImageNet classification with T2T-ViT, KDEformer shows over $18\times$ speedup while the accuracy drop is less than $0.5\%$.

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.

This paper investigates the post-hoc calibration of confidence for "exploratory" machine learning classification problems. The difficulty in these problems stems from the continuing desire to push the boundaries of which categories have enough examples to generalize from when curating datasets, and confusion regarding the validity of those categories. We argue that for such problems the "one-versus-all" approach (top-label calibration) must be used rather than the "calibrate-the-full-response-matrix" approach advocated elsewhere in the literature. We introduce and test four new algorithms designed to handle the idiosyncrasies of category-specific confidence estimation. Chief among these methods is the use of kernel density ratios for confidence calibration including a novel, bulletproof algorithm for choosing the bandwidth. We test our claims and explore the limits of calibration on a bioinformatics application (PhANNs) as well as the classic MNIST benchmark. Finally, our analysis argues that post-hoc calibration should always be performed, should be based only on the test dataset, and should be sanity-checked visually.

Estimation of the value-at-risk (VaR) of a large portfolio of assets is an important task for financial institutions. As the joint log-returns of asset prices can often be projected to a latent space of a much smaller dimension, the use of a variational autoencoder (VAE) for estimating the VaR is a natural suggestion. To ensure the bottleneck structure of autoencoders when learning sequential data, we use a temporal VAE (TempVAE) that avoids an auto-regressive structure for the observation variables. However, the low signal- to-noise ratio of financial data in combination with the auto-pruning property of a VAE typically makes the use of a VAE prone to posterior collapse. Therefore, we propose to use annealing of the regularization to mitigate this effect. As a result, the auto-pruning of the TempVAE works properly which also results in excellent estimation results for the VaR that beats classical GARCH-type and historical simulation approaches when applied to real data.

State density distribution, in contrast to worst-case reachability, can be leveraged for safety-related problems to better quantify the likelihood of the risk for potentially hazardous situations. In this work, we propose a data-driven method to compute the density distribution of reachable states for nonlinear and even black-box systems. Our semi-supervised approach learns system dynamics and the state density jointly from trajectory data, guided by the fact that the state density evolution follows the Liouville partial differential equation. With the help of neural network reachability tools, our approach can estimate the set of all possible future states as well as their density. Moreover, we could perform online safety verification with probability ranges for unsafe behaviors to occur. We use an extensive set of experiments to show that our learned solution can produce a much more accurate estimate on density distribution, and can quantify risks less conservatively and flexibly comparing with worst-case analysis.

Designing functional molecules and advanced materials requires complex interdependent design choices: tuning continuous process parameters such as temperatures or flow rates, while simultaneously selecting categorical variables like catalysts or solvents. To date, the development of data-driven experiment planning strategies for autonomous experimentation has largely focused on continuous process parameters despite the urge to devise efficient strategies for the selection of categorical variables to substantially accelerate scientific discovery. We introduce Gryffin, as a general purpose optimization framework for the autonomous selection of categorical variables driven by expert knowledge. Gryffin augments Bayesian optimization with kernel density estimation using smooth approximations to categorical distributions. Leveraging domain knowledge from physicochemical descriptors to characterize categorical options, Gryffin can significantly accelerate the search for promising molecules and materials. Gryffin can further highlight relevant correlations between the provided descriptors to inspire physical insights and foster scientific intuition. In addition to comprehensive benchmarks, we demonstrate the capabilities and performance of Gryffin on three examples in materials science and chemistry: (i) the discovery of non-fullerene acceptors for organic solar cells, (ii) the design of hybrid organic-inorganic perovskites for light-harvesting, and (iii) the identification of ligands and process parameters for Suzuki-Miyaura reactions. Our observations suggest that Gryffin, in its simplest form without descriptors, constitutes a competitive categorical optimizer compared to state-of-the-art approaches. However, when leveraging domain knowledge provided via descriptors, Gryffin can optimize at considerable higher rates and refine this domain knowledge to spark scientific understanding.