Adversarial Lagrangian Integrated Contrastive Embedding for Limited Size Datasets

Certain datasets contain a limited number of samples with highly various styles and complex structures. This study presents a novel adversarial Lagrangian integrated contrastive embedding (ALICE) method for small-sized datasets. First, the accuracy improvement and training convergence of the proposed pre-trained adversarial transfer are shown on various subsets of datasets with few samples. Second, a novel adversarial integrated contrastive model using various augmentation techniques is investigated. The proposed structure considers the input samples with different appearances and generates a superior representation with adversarial transfer contrastive training. Finally, multi-objective augmented Lagrangian multipliers encourage the low-rank and sparsity of the presented adversarial contrastive embedding to adaptively estimate the coefficients of the regularizers automatically to the optimum weights. The sparsity constraint suppresses less representative elements in the feature space. The low-rank constraint eliminates trivial and redundant components and enables superior generalization. The performance of the proposed model is verified by conducting ablation studies by using benchmark datasets for scenarios with small data samples.

Object Type Clustering using Markov Directly-Follow Multigraph in Object-Centric Process Mining

Object-centric process mining is a new paradigm with more realistic assumptions about underlying data by considering several case notions, e.g., an order handling process can be analyzed based on order, item, package, and route case notions. Including many case notions can result in a very complex model. To cope with such complexity, this paper introduces a new approach to cluster similar case notions based on Markov Directly-Follow Multigraph, which is an extended version of the well-known Directly-Follow Graph supported by many industrial and academic process mining tools. This graph is used to calculate a similarity matrix for discovering clusters of similar case notions based on a threshold. A threshold tuning algorithm is also defined to identify sets of different clusters that can be discovered based on different levels of similarity. Thus, the cluster discovery will not rely on merely analysts' assumptions. The approach is implemented and released as a part of a python library, called processmining, and it is evaluated through a Purchase to Pay (P2P) object-centric event log file. Some discovered clusters are evaluated by discovering Directly Follow-Multigraph by flattening the log based on the clusters. The similarity between identified clusters is also evaluated by calculating the similarity between the behavior of the process models discovered for each case notion using inductive miner based on footprints conformance checking.

Evaluating Perceived Usefulness and Ease of Use of CMMN and DCR

Case Management has been gradually evolving to support Knowledge-intensive business process management, which resulted in developing different modeling languages, e.g., Declare, Dynamic Condition Response (DCR), and Case Management Model and Notation (CMMN). A language will die if users do not accept and use it in practice - similar to extinct human languages. Thus, it is important to evaluate how users perceive languages to determine if there is a need for improvement. Although some studies have investigated how the process designers perceived Declare and DCR, there is a lack of research on how they perceive CMMN. Therefore, this study investigates how the process designers perceive the usefulness and ease of use of CMMN and DCR based on the Technology Acceptance Model. DCR is included to enable comparing the study result with previous ones. The study is performed by educating master level students with these languages over eight weeks by giving feedback on their assignments to reduce perceptions biases. The students' perceptions are collected through questionnaires before and after sending feedback on their final practice in the exam. Thus, the result shows how the perception of participants can change by receiving feedback - despite being well trained. The reliability of responses is tested using Cronbach's alpha, and the result indicates that both languages have an acceptable level for both perceived usefulness and ease of use.

Persistent Reductions in Regularized Loss Minimization for Variable Selection

In the context of regularized loss minimization with polyhedral gauges, we show that for a broad class of loss functions (possibly non-smooth and non-convex) and under a simple geometric condition on the input data it is possible to efficiently identify a subset of features which are guaranteed to have zero coefficients in all optimal solutions in all problems with loss functions from said class, before any iterative optimization has been performed for the original problem. This procedure is standalone, takes only the data as input, and does not require any calls to the loss function. Therefore, we term this procedure as a persistent reduction for the aforementioned class of regularized loss minimization problems. This reduction can be efficiently implemented via an extreme ray identification subroutine applied to a polyhedral cone formed from the datapoints. We employ an existing output-sensitive algorithm for extreme ray identification which makes our guarantee and algorithm applicable in ultra-high dimensional problems.

New Computational and Statistical Aspects of Regularized Regression with Application to Rare Feature Selection and Aggregation

Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying projection oracle in this case and how it can be efficiently represented as a quadratically constrained quadratic program. Our motivation for the study of this norm is regularized regression in the presence of rare features which poses a challenge to various methods within high-dimensional statistics, and in machine learning in general. The proposed estimation procedure is designed to perform automatic feature selection and aggregation for which we develop statistical bounds. The bounds are general and offer a statistical framework for norm-based regularization. The bounds rely on novel geometric quantities on which we attempt to elaborate as well.

2-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANs

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator.

Missing Data in Sparse Transition Matrix Estimation for Sub-Gaussian Vector Autoregressive Processes

High-dimensional time series data exist in numerous areas such as finance, genomics, healthcare, and neuroscience. An unavoidable aspect of all such datasets is missing data, and dealing with this issue has been an important focus in statistics, control, and machine learning. In this work, we consider a high-dimensional estimation problem where a dynamical system, governed by a stable vector autoregressive model, is randomly and only partially observed at each time point. Our task amounts to estimating the transition matrix, which is assumed to be sparse. In such a scenario, where covariates are highly interdependent and partially missing, new theoretical challenges arise. While transition matrix estimation in vector autoregressive models has been studied previously, the missing data scenario requires separate efforts. Moreover, while transition matrix estimation can be studied from a high-dimensional sparse linear regression perspective, the covariates are highly dependent and existing results on regularized estimation with missing data from i.i.d.~covariates are not applicable. At the heart of our analysis lies 1) a novel concentration result when the innovation noise satisfies the convex concentration property, as well as 2) a new quantity for characterizing the interactions of the time-varying observation process with the underlying dynamical system.

Subspace Clustering with Missing and Corrupted Data

Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a weighted combination of the other samples, with weights of minimal $\ell_1$ norm, and then uses those learned weights to cluster the samples. SSC is stable in settings where each sample is contaminated by a relatively small amount of noise. However, when there is a significant amount of additive noise, or a considerable number of entries are missing, theoretical guarantees are scarce. In this paper, we study a robust variant of SSC and establish clustering guarantees in the presence of corrupted or missing data. We give explicit bounds on amount of noise and missing data that the algorithm can tolerate, both in deterministic settings and in a random generative model. Notably, our approach provides guarantees for higher tolerance to noise and missing data than existing analyses for this method. By design, the results hold even when we do not know the locations of the missing data; e.g., as in presence-only data.

Variational Gram Functions: Convex Analysis and Optimization

We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, multitask learning, and estimating vectors with disjoint supports, among other applications. We study convexity for VGFs, and give efficient characterizations for their convex conjugates, subdifferentials, and proximal operators. We discuss efficient optimization algorithms for regularized loss minimization problems where the loss admits a common, yet simple, variational representation and the regularizer is a VGF. These algorithms enjoy a simple kernel trick, an efficient line search, as well as computational advantages over first order methods based on the subdifferential or proximal maps. We also establish a general representer theorem for such learning problems. Lastly, numerical experiments on a hierarchical classification problem are presented to demonstrate the effectiveness of VGFs and the associated optimization algorithms.