Enabling Explainable Fusion in Deep Learning with Fuzzy Integral Neural Networks

Information fusion is an essential part of numerous engineering systems and biological functions, e.g., human cognition. Fusion occurs at many levels, ranging from the low-level combination of signals to the high-level aggregation of heterogeneous decision-making processes. While the last decade has witnessed an explosion of research in deep learning, fusion in neural networks has not observed the same revolution. Specifically, most neural fusion approaches are ad hoc, are not understood, are distributed versus localized, and/or explainability is low (if present at all). Herein, we prove that the fuzzy Choquet integral (ChI), a powerful nonlinear aggregation function, can be represented as a multi-layer network, referred to hereafter as ChIMP. We also put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient descent-based optimization in light of the exponential number of ChI inequality constraints. An additional benefit of ChIMP/iChIMP is that it enables eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP is applied to the fusion of a set of heterogeneous architecture deep models in remote sensing. We show an improvement in model accuracy and our previously established XAI indices shed light on the quality of our data, model, and its decisions.Comment: IEEE Transactions on Fuzzy System

Efficient Data Driven Multi Source Fusion

Data/information fusion is an integral component of many existing and emerging applications; e.g., remote sensing, smart cars, Internet of Things (IoT), and Big Data, to name a few. While fusion aims to achieve better results than what any one individual input can provide, often the challenge is to determine the underlying mathematics for aggregation suitable for an application. In this dissertation, I focus on the following three aspects of aggregation: (i) efficient data-driven learning and optimization, (ii) extensions and new aggregation methods, and (iii) feature and decision level fusion for machine learning with applications to signal and image processing. The Choquet integral (ChI), a powerful nonlinear aggregation operator, is a parametric way (with respect to the fuzzy measure (FM)) to generate a wealth of aggregation operators. The FM has 2N variables and N(2N − 1) constraints for N inputs. As a result, learning the ChI parameters from data quickly becomes impractical for most applications. Herein, I propose a scalable learning procedure (which is linear with respect to training sample size) for the ChI that identifies and optimizes only data-supported variables. As such, the computational complexity of the learning algorithm is proportional to the complexity of the solver used. This method also includes an imputation framework to obtain scalar values for data-unsupported (aka missing) variables and a compression algorithm (lossy or losselss) of the learned variables. I also propose a genetic algorithm (GA) to optimize the ChI for non-convex, multi-modal, and/or analytical objective functions. This algorithm introduces two operators that automatically preserve the constraints; therefore there is no need to explicitly enforce the constraints as is required by traditional GA algorithms. In addition, this algorithm provides an efficient representation of the search space with the minimal set of vertices. Furthermore, I study different strategies for extending the fuzzy integral for missing data and I propose a GOAL programming framework to aggregate inputs from heterogeneous sources for the ChI learning. Last, my work in remote sensing involves visual clustering based band group selection and Lp-norm multiple kernel learning based feature level fusion in hyperspectral image processing to enhance pixel level classification

Feature and Decision Level Fusion Using Multiple Kernel Learning and Fuzzy Integrals

The work collected in this dissertation addresses the problem of data fusion. In other words, this is the problem of making decisions (also known as the problem of classification in the machine learning and statistics communities) when data from multiple sources are available, or when decisions/confidence levels from a panel of decision-makers are accessible. This problem has become increasingly important in recent years, especially with the ever-increasing popularity of autonomous systems outfitted with suites of sensors and the dawn of the ``age of big data.\u27\u27 While data fusion is a very broad topic, the work in this dissertation considers two very specific techniques: feature-level fusion and decision-level fusion. In general, the fusion methods proposed throughout this dissertation rely on kernel methods and fuzzy integrals. Both are very powerful tools, however, they also come with challenges, some of which are summarized below. I address these challenges in this dissertation. Kernel methods for classification is a well-studied area in which data are implicitly mapped from a lower-dimensional space to a higher-dimensional space to improve classification accuracy. However, for most kernel methods, one must still choose a kernel to use for the problem. Since there is, in general, no way of knowing which kernel is the best, multiple kernel learning (MKL) is a technique used to learn the aggregation of a set of valid kernels into a single (ideally) superior kernel. The aggregation can be done using weighted sums of the pre-computed kernels, but determining the summation weights is not a trivial task. Furthermore, MKL does not work well with large datasets because of limited storage space and prediction speed. These challenges are tackled by the introduction of many new algorithms in the following chapters. I also address MKL\u27s storage and speed drawbacks, allowing MKL-based techniques to be applied to big data efficiently. Some algorithms in this work are based on the Choquet fuzzy integral, a powerful nonlinear aggregation operator parameterized by the fuzzy measure (FM). These decision-level fusion algorithms learn a fuzzy measure by minimizing a sum of squared error (SSE) criterion based on a set of training data. The flexibility of the Choquet integral comes with a cost, however---given a set of N decision makers, the size of the FM the algorithm must learn is 2N. This means that the training data must be diverse enough to include 2N independent observations, though this is rarely encountered in practice. I address this in the following chapters via many different regularization functions, a popular technique in machine learning and statistics used to prevent overfitting and increase model generalization. Finally, it is worth noting that the aggregation behavior of the Choquet integral is not intuitive. I tackle this by proposing a quantitative visualization strategy allowing the FM and Choquet integral behavior to be shown simultaneously

Comparison of Fuzzy Integral-Fuzzy Measure based Ensemble Algorithms with the State-of-the-art Ensemble Algorithms

The Fuzzy Integral (FI) is a non-linear aggregation operator which enables the fusion of information from multiple sources in respect to a Fuzzy Measure (FM) which captures the worth of both the individual sources and all their possible combinations. Based on the expected potential of non-linear aggregation offered by the FI, its application to decision-level fusion in ensemble classifiers, i.e. to fuse multiple classifiers outputs towards one superior decision level output, has recently been explored. A key example of such a FI-FM ensemble classification method is the Decision-level Fuzzy Integral Multiple Kernel Learning (DeFIMKL) algorithm, which aggregates the outputs of kernel based classifiers through the use of the Choquet FI with respect to a FM learned through a regularised quadratic programming approach. While the approach has been validated against a number of classifiers based on multiple kernel learning, it has thus far not been compared to the state-of-the-art in ensemble classification. Thus, this paper puts forward a detailed comparison of FI-FM based ensemble methods, specifically the DeFIMKL algorithm, with state-of-the art ensemble methods including Adaboost, Bagging, Random Forest and Majority Voting over 20 public datasets from the UCI machine learning repository. The results on the selected datasets suggest that the FI based ensemble classifier performs both well and efficiently, indicating that it is a viable alternative when selecting ensemble classifiers and indicating that the non-linear fusion of decision level outputs offered by the FI provides expected potential and warrants further study

Insights and Characterization of l1-norm Based Sparsity Learning of a Lexicographically Encoded Capacity Vector for the Choquet Integral

This thesis aims to simultaneously minimize function error and model complexity for data fusion via the Choquet integral (CI). The CI is a generator function, i.e., it is parametric and yields a wealth of aggregation operators based on the specifics of the underlying fuzzy measure. It is often the case that we desire to learn a fusion from data and the goal is to have the smallest possible sum of squared error between the trained model and a set of labels. However, we also desire to learn as “simple’’ of solutions as possible. Herein, L1-norm regularization of a lexicographically encoded capacity vector relative to the CI is explored. The impact of regularization is explored in terms of what capacities and aggregation operators it induces under different common and extreme scenarios. Synthetic experiments are provided in order to illustrate the propositions and concepts put forth

Multiple Instance Choquet Integral for multiresolution sensor fusion

Imagine you are traveling to Columbia, MO for the first time. On your flight to Columbia, the woman sitting next to you recommended a bakery by a large park with a big yellow umbrella outside. After you land, you need directions to the hotel from the airport. Suppose you are driving a rental car, you will need to park your car at a parking lot or a parking structure. After a good night's sleep in the hotel, you may decide to go for a run in the morning on the closest trail and stop by that recommended bakery under a big yellow umbrella. It would be helpful in the course of completing all these tasks to accurately distinguish the proper car route and walking trail, find a parking lot, and pinpoint the yellow umbrella. Satellite imagery and other geo-tagged data such as Open Street Maps provide effective information for this goal. Open Street Maps can provide road information and suggest bakery within a five-mile radius. The yellow umbrella is a distinctive color and, perhaps, is made of a distinctive material that can be identified from a hyperspectral camera. Open Street Maps polygons are tagged with information such as "parking lot" and "sidewalk." All these information can and should be fused to help identify and offer better guidance on the tasks you are completing. Supervised learning methods generally require precise labels for each training data point. It is hard (and probably at an extra cost) to manually go through and label each pixel in the training imagery. GPS coordinates cannot always be fully trusted as a GPS device may only be accurate to the level of several pixels. In many cases, it is practically infeasible to obtain accurate pixel-level training labels to perform fusion for all the imagery and maps available. Besides, the training data may come in a variety of data types, such as imagery or as a 3D point cloud. The imagery may have different resolutions, scales and, even, coordinate systems. Previous fusion methods are generally only limited to data mapped to the same pixel grid, with accurate labels. Furthermore, most fusion methods are restricted to only two sources, even if certain methods, such as pan-sharpening, can deal with different geo-spatial types or data of different resolution. It is, therefore, necessary and important, to come up with a way to perform fusion on multiple sources of imagery and map data, possibly with different resolutions and of different geo-spatial types with consideration of uncertain labels. I propose a Multiple Instance Choquet Integral framework for multi-resolution multisensor fusion with uncertain training labels. The Multiple Instance Choquet Integral (MICI) framework addresses uncertain training labels and performs both classification and regression. Three classifier fusion models, i.e. the noisy-or, min-max, and generalized-mean models, are derived under MICI. The Multi-Resolution Multiple Instance Choquet Integral (MR-MICI) framework is built upon the MICI framework and further addresses multiresolution in the fusion sources in addition to the uncertainty in training labels. For both MICI and MR-MICI, a monotonic normalized fuzzy measure is learned to be used with the Choquet integral to perform two-class classifier fusion given bag-level training labels. An optimization scheme based on the evolutionary algorithm is used to optimize the models proposed. For regression problems where the desired prediction is real-valued, the primary instance assumption is adopted. The algorithms are applied to target detection, regression and scene understanding applications. Experiments are conducted on the fusion of remote sensing data (hyperspectral and LiDAR) over the campus of University of Southern Mississippi - Gulfpark. Clothpanel sub-pixel and super-pixel targets were placed on campus with varying levels of occlusion and the proposed algorithms can successfully detect the targets in the scene. A semi-supervised approach is developed to automatically generate training labels based on data from Google Maps, Google Earth and Open Street Map. Based on such training labels with uncertainty, the proposed algorithms can also identify materials on campus for scene understanding, such as road, buildings, sidewalks, etc. In addition, the algorithms are used for weed detection and real-valued crop yield prediction experiments based on remote sensing data that can provide information for agricultural applications.Includes biblographical reference

Extension of the fuzzy integral for general fuzzy set-valued information

The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multicriteria decision making, skeletal age-at-death estimation, and multisource (e.g., feature, algorithm, sensor, and confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number (FN), i.e., convex and normal fuzzy set- (FS) valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non-EP-based FI extension called the nondirect FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multisource fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications

EXPLAINABLE FEATURE- AND DECISION-LEVEL FUSION

Information fusion is the process of aggregating knowledge from multiple data sources to produce more consistent, accurate, and useful information than any one individual source can provide. In general, there are three primary sources of data/information: humans, algorithms, and sensors. Typically, objective data---e.g., measurements---arise from sensors. Using these data sources, applications such as computer vision and remote sensing have long been applying fusion at different levels (signal, feature, decision, etc.). Furthermore, the daily advancement in engineering technologies like smart cars, which operate in complex and dynamic environments using multiple sensors, are raising both the demand for and complexity of fusion. There is a great need to discover new theories to combine and analyze heterogeneous data arising from one or more sources. The work collected in this dissertation addresses the problem of feature- and decision-level fusion. Specifically, this work focuses on fuzzy choquet integral (ChI)-based data fusion methods. Most mathematical approaches for data fusion have focused on combining inputs relative to the assumption of independence between them. However, often there are rich interactions (e.g., correlations) between inputs that should be exploited. The ChI is a powerful aggregation tool that is capable modeling these interactions. Consider the fusion of m sources, where there are 2m unique subsets (interactions); the ChI is capable of learning the worth of each of these possible source subsets. However, the complexity of fuzzy integral-based methods grows quickly, as the number of trainable parameters for the fusion of m sources scales as 2m. Hence, we require a large amount of training data to avoid the problem of over-fitting. This work addresses the over-fitting problem of ChI-based data fusion with novel regularization strategies. These regularization strategies alleviate the issue of over-fitting while training with limited data and also enable the user to consciously push the learned methods to take a predefined, or perhaps known, structure. Also, the existing methods for training the ChI for decision- and feature-level data fusion involve quadratic programming (QP). The QP-based learning approach for learning ChI-based data fusion solutions has a high space complexity. This has limited the practical application of ChI-based data fusion methods to six or fewer input sources. To address the space complexity issue, this work introduces an online training algorithm for learning ChI. The online method is an iterative gradient descent approach that processes one observation at a time, enabling the applicability of ChI-based data fusion on higher dimensional data sets. In many real-world data fusion applications, it is imperative to have an explanation or interpretation. This may include providing information on what was learned, what is the worth of individual sources, why a decision was reached, what evidence process(es) were used, and what confidence does the system have on its decision. However, most existing machine learning solutions for data fusion are black boxes, e.g., deep learning. In this work, we designed methods and metrics that help with answering these questions of interpretation, and we also developed visualization methods that help users better understand the machine learning solution and its behavior for different instances of data

Conceptual Design Evaluation of Mechatronic Systems

The definition of the conceptual design phase has been expressed in many different phrasings, but all of them lead to the same conclusion. The conceptual design phase is of the highest importance during the design process, due to the fact that many crucial decisions concerning the progress of the design need to be taken with very little to none information and knowledge about the design object. This implies to very high uncertainty about the effects that these decisions will have later on. During the conceptual design of a mechatronic system, the system to be designed is modeled, and several solutions (alternatives) to the design problem are generated and evaluated so that the most fitting one to the design specifications and requirements is chosen. The purpose of this chapter is to mention some of the most widely used methods of system modeling, mainly through hierarchical representations of their subsystems, and also to present a method for the generation and evaluation of the design alternatives