The Choquet integral (ChI) is an aggregation operator defined with respect to a fuzzy measure (FM). The FM encodes the worth of all subsets of the sources of information that are being aggregated. The ChI is capable of representing many aggregation functions and has found its application in a wide range of decision fusion problems. In our prior work, we introduced a data support-based approach for learning the FM for decision fusion problems. This approach applies a quadratic programming (QP)-based method to train the FM. However, since the FM of ChI scales as$2^{N}$, where$N$is the number of input sources, the space complexity for learning the FM grows exponentially with$N$. This has limited the practical application of ChI-based decision fusion methods to small numbers of dimenstions—$N$≲ 6 is practical in most cases. In this work, we propose an iterative gradient descent-based approach to train the FM for ChI with an efficient method for handling the FM constraints. This method processes the training data, one observation at a time, and thereby significantly reduces the space complexity of the training process. We tested our online method on synthetic and real-world data sets, and compared the performance and convergence behaviour with our previously proposed QP-based method (i.e., batch method). On 10 out of 12 data sets, the online learning method has either matched or outperformed the batch method. We also show that we are able to use larger numbers of inputs with the online learning approach, extending the practical application of the ChI.

Fuzzy integrals (FIs) are powerful aggregation operators that fuse information from multiple sources. The aggregation is parameterized using a fuzzy measure (FM), which encodes the worths of all subsets of sources. Since the FI is defined with respect to an FM, much consideration must be given to defining the FM. However, in practice this is a difficult task—the number of values in an FM scales as$2^n$, where$n$is the number of input sources, thus manually specifying an FM quickly becomes tedious. In this article, we review an automatic, data-supported method of learning the FM by minimizing a sum-of-squared error objective function in the context of decision-level fusion of classifiers using the Choquet FI. While this solves the specification problem, we illuminate an issue encountered with many real-world data sets; i.e., if the training data do not contain a significant number of all possible sort orders, many of the FM values are not supported by the data. We propose various regularization strategies to alleviate this issue by pushing the learned FM toward a predefined structure; these regularizers allow the user to encode knowledge of the underlying FM to the learning problem. Furthermore, we propose another regularization strategy that constrains the learned FM's structure to be a linear order statistic. Finally, we perform several experiments using synthetic and real-world data sets and show that our proposed extensions can improve the learned FM behavior and classification accuracy. A previously proposed visualization technique is employed to simultaneously quantitatively illustrate the FM as well as the FI.

The ordered weighted average (OWA) operator is a well-known aggregation tool that is primarily used for decisionlevel fusion. However, the OWA is a convex sum, i.e., its learned coefficients are constrained to sum to one, and thus the output is restricted to lie between the maximum and minimum values of the inputs. Relaxing this constraint on the sum of weights transforms the OWA into a linear order statistic (LOS), which allows the aggregation operation to map the input to any value on the set of reals, thus behaving more like a regression operator. The LOS parameterizes the regression operation of d-features using just d parameters, which helps with the model's interpretability. However, learning just d parameters limits the amount of nonlinear space explored for an optimal solution, and thus reduces the expressibility of the LOS algorithm. We propose a novel aggregation method called the extended linear order statistic (ELOS), where for each position in the sorted input vector we have d parameters, one for each input feature, thus learning a total of d2weights for the aggregation of d features. The increased number of parameters helps the algorithm improve its expressibility while maintaining interpretability. In our experiments on real-world benchmark data sets, ELOS has outperformed both linear regression and LOS in 8 out of 10 experiments.

The Choquet integral (ChI) is an aggregation function that is defined with respect to a fuzzy measure (FM). Many ChI-based decision aggregation methods have been proposed to learn the underlying FM. However, FM's boundary and monotonicity constraints have limited the applicability of such methods to decision-level fusion. In a recent work, we removed the constraints on FM to develop a regression model based on the ChI. Our model has a generalized bias that enables capability beyond previously proposed ChI regression approaches. We also developed an approach for learning the parameters of the ChI regression from training data. In this paper, we develop a method to apply ℓ2-regularization on our training algorithm. In our experiments on real-world benchmark data sets, ridge regularized-ChI regression has outperformed the unregularized version in 22 out of 30 (73%) experiments. Also, when compared with several competing regression methods, results show that our approach has superior performance.

The Choquet Integral (ChI) is an aggregation operator defined with respect to a Fuzzy Measure (FM). The FM encodes the worth of all subsets of the sources of information that are being aggregated. The monotonicity and the boundary conditions of the FM have limited its applicability to decision-level fusion. But in a recent work, we removed the boundary and monotonicity constraints of the FM, which we then called a bounded capacity (BC), to propose a Choquet Integral regression (CIR) approach that enables capability beyond previously proposed ChI regression methods. In the same work, we also presented a quadratic programming (QP)-based method, batch-CIR, to learn the BC parameters of the CIR from training data. However, the QP used for learning the BC scales exponentially with the dimensionality of the training data and thus it becomes impractical on data sets with 7 or more dimensions. In this paper we propose an iterative gradient descent approach, online-CIR, to learn the BC. This method iteratively processes the training data, one data point at a time, and therefore requires significantly less computation and space at any time during the training. The application of batch-CIR required the dimensionality reduction of high-dimensional data sets to enable computation in a reasonable time. The proposed online-CIR approach has enabled us to extend CIR to data sets with larger dimensionality. In our experimental evaluation using benchmark regression data sets, online-CIR has outperformed batch-CIR on high-dimensional data sets while also matching the batch-CIR performance on low-dimensional data sets.

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 multilayer 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 artificial intelligence (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.

The Choquet integral (ChI) is a proven tool for information aggregation. In prior work, we showed that learning a ChI from data results in missing variables. Herein, we explore two ways to transfer a known ChI from a source domain to a new under sampled target domain. The first method is based on regularization and it listens to the full source domain ChI. The second method optimizes what we can observe (target domain supported variables) and missing variables are the only thing migrated from the source domain. Synthetic experiments, aka we know the truth, are used to show the behavior of these methods with respect to transfering between ChIs.

Information or data aggregation is an important part of nearly all analysis problems as summarizing inputs from multiple sources is a ubiquitous goal. In this paper we propose a method for non-linear aggregation of data inputs that take the form of non-normal fuzzy sets. The proposed shape-preserving fuzzy integral (SPFI) is designed to overcome a well-known weakness of the previously-proposed sub-normal fuzzy integral (SuFI). The weakness of SuFI is that the output is constrained to have maximum membership equal to the minimum of the maximum memberships of the inputs; hence, if one input has a small height, then the output is constrained to that height. The proposed SPFI does not suffer from this weakness and, furthermore, preserves in the output the shape of the input sets. That is, the output looks like the inputs. The SPFI method is based on the well-known Choquet fuzzy integral with respect to a capacity measure, i.e., fuzzy measure. We demonstrate SPFI on synthetic and real-world data, comparing it to the SuFI and non-direct fuzzy integral (NDFI).

To date, numerous ways have been created to learn a fusion solution from data. However, a gap exists in terms of understanding the quality of what was learned and how trustworthy the fusion is for future-i.e., new-data. In part, the current paper is driven by the demand for so-called explainable AI (XAI). Herein, we discuss methods for XAI of the Choquet integral (ChI), a parametric nonlinear aggregation function. Specifically, we review existing indices, and we introduce new data-centric XAI tools. These various XAI-ChI methods are explored in the context of fusing a set of heterogeneous deep convolutional neural networks for remote sensing.

The Choquet integral (ChI) is a parametric nonlinear aggregation function defined with respect to the fuzzy measure (FM). To date, application of the ChI has sadly been restricted to problems with relatively few numbers of inputs; primarily as the FM has 2Nvariables for N inputs and N(2N-1- 1) monotonicity constraints. In return, the community has turned to density-based imputation (e.g., Sugeno λ-FM) or the number of interactions (FM variables) are restricted (e.g., k-additivity). Herein, we propose a new scalable data-driven way to represent and learn the ChI, making learning computationally manageable for larger N. First, data supported variables are identified and used in optimization. Identification of these variables also allows us recognize future ill-posed fusion scenarios; ChIs involving variable subsets not supported by data. Second, we outline an imputation function framework to address data unsupported variables. Third, we present a lossless way to compress redundant variables and associated monotonicity constraints. Finally, we outline a lossy approximation method to further compress the ChI (if/when desired). Computational complexity analysis and experiments conducted on synthetic datasets with known FMs demonstrate the effectiveness and efficiency of the proposed theory.

The fuzzy integral (FI) is a nonlinear aggregation operator whose behavior is defined by the fuzzy measure (FM). As an aggregation operator, the FI is commonly used for evidence fusion where it combines sources of information based on the worth of each subset of sources. One drawback to FI-based methods, however, is the specification of the FM. Defining the FM manually quickly becomes too tedious since the number of FM terms scales as 2n, where n is the number of sources; thus, an automatic method of defining the FM is necessary. In this paper, we review a data-driven method of learning the FM via minimizing the sum-of-squared error (SSE) in the context of decision-level fusion and propose an extension allowing knowledge of the underlying FM to be encoded in the algorithm. The algorithm is applied to real-world and toy datasets and results show that the extension can improve classification accuracy. Furthermore, we introduce a visualization strategy to simultaneously show the quantitative information in the FM as well as the FI.

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 precomputed 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. In this paper, we address all three of these multiple kernel challenges. First, we introduce a new linear feature level fusion technique and learning algorithm, GAMKLp. Second, we put forth three new algorithms, DeFIMKL, DeGAMKL, and DeLSMKL, for nonlinear fusion of kernels at the decision level. To address MKL's storage and speed drawbacks, we apply the Nystrom approximation to the kernel matrices. We compare our methods to a successful and state-of-the-art technique called MKL-group lasso (MKLGL), and experiments on several benchmark datasets show that some of our proposed algorithms outperform MKLGL when applied to support vector machine (SVM)-based classification. However, to no surprise, there does not seem to be a global winner but instead different strategies that a user can employ. Experiments with our kernel approximation method show that we can routinely discard most of the training data and at least double prediction speed without sacrificing classification accuracy. These results suggest that MKL-based classification techniques can be applied to big data efficiently, which is confirmed by an experiment using a large dataset.

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. A popular and successful approach to this problem is MKL-group lasso (MKLGL), where the weights and classification surface are simultaneously solved by iteratively optimizing a min-max optimization until convergence. In this work, we propose an ℓp-normed genetic algorithm MKL (GAMKLp), which uses a genetic algorithm to learn the weights of a set of pre-computed kernel matrices for use with MKL classification. We prove that this approach is equivalent to a previously proposed fuzzy integral aggregation of multiple kernels called fuzzy integral: genetic algorithm (FIGA). A second algorithm, which we call decision-level fuzzy integral MKL (DeFIMKL), is also proposed, where a fuzzy measure with respect to the fuzzy Choquet integral is learned via quadratic programming, and the decision value-viz., the class label-is computed using the fuzzy Choquet integral aggregation. Experiments on several benchmark data sets show that our proposed algorithms can outperform MKLGL when applied to support vector machine (SVM)-based classification.

An underwater glider named ROUGHIE (Research Oriented Underwater Glider for Hands-on Investigative Engineering) is designed and manufactured to provide a test platform and framework for experimental underwater automation. This paper presents an efficient multi-level motion controller that can be used to enhance underwater glider control systems or easily modified for additional sensing, computing, or other requirements for advanced automation design testing. The ultimate goal is to have a fleet of modular and inexpensive test platforms for addressing the issues that currently limit the use of autonomous underwater vehicles (AUVs). Producing a low-cost vehicle with maneuvering capabilities and a straightforward expansion path will permit easy experimentation and testing of different approaches to improve underwater automation.