Generative adversarial networks (GANs) are deep neural networks that allow us to sample from an arbitrary probability distribution without explicitly estimating the distribution. There is a generator that takes a latent vector as input and transforms it into a valid sample from the distribution. There is also a discriminator that is trained to discriminate such fake samples from true samples of the distribution; at the same time, the generator is trained to generate fakes that the discriminator cannot tell apart from the true samples. Instead of learning a global generator, a recent approach involves training multiple generators each responsible from one part of the distribution. In this work, we review such approaches and propose the hierarchical mixture of generators, inspired from the hierarchical mixture of experts model, that learns a tree structure implementing a hierarchical clustering with soft splits in the decision nodes and local generators in the leaves. Since the generators are combined softly, the whole model is continuous and can be trained using gradient-based optimization, just like the original GAN model. Our experiments on five image data sets, namely, MNIST, FashionMNIST, UTZap50K, Oxford Flowers, and CelebA, show that our proposed model generates samples of high quality and diversity in terms of popular GAN evaluation metrics. The learned hierarchical structure also leads to knowledge extraction.

Traditionally, deep learning algorithms update the network weights, whereas the network architecture is chosen manually using a process of trial and error. In this paper, we propose two novel approaches that automatically update the network structure while also learning its weights. The novelty of our approach lies in our parameterization, where the depth, or additional complexity, is encapsulated continuously in the parameter space through control parameters that add additional complexity. We propose two methods. In tunnel networks, this selection is done at the level of a hidden unit, and in budding perceptrons, this is done at the level of a network layer; updating this control parameter introduces either another hidden unit or layer. We show the effectiveness of our methods on the synthetic two-spiral data and on three real data sets of MNIST, MIRFLICKR, and CIFAR, where we see that our proposed methods, with the same set of hyperparameters, can correctly adjust the network complexity to the task complexity.

Error Correcting Output Coding (ECOC) is a multiclass classification technique in which multiple binary classifiers are trained according to a preset code matrix such that each one learns a separate dichotomy of the classes. While ECOC is one of the best solutions for multi-class problems, one issue which makes it suboptimal is that the training of the base classifiers is done independently of the generation of the code matrix. In this paper, we propose to modify a given ECOC matrix to improve its performance by reducing this decoupling. The proposed algorithm uses beam search to iteratively modify the original matrix, using validation accuracy as a guide. It does not involve further training of the classifiers and can be applied to any ECOC matrix. We evaluate the accuracy of the proposed algorithm (BeamECOC) using 10-fold cross-validation experiments on 6 UCI datasets, using random code matrices of different sizes, and base classifiers of different strengths. Compared to the random ECOC approach, BeamECOC increases the average cross-validation accuracy in 83.3% of the experimental settings involving all datasets, and gives better results than the state-of-the-art in 75% of the scenarios. By employing BeamECOC, it is also possible to reduce the number of columns of a random matrix down to 13% and still obtain comparable or even better results at times.

We propose a new decision tree model, named the budding tree, where a node can be both a leaf and an internal decision node. Each bud node starts as a leaf node, can then grow children, but then later on, if necessary, its children can be pruned. This contrasts with traditional tree construction algorithms that only grows the tree during the training phase, and prunes it in a separate pruning phase. We use a soft tree architecture and show that the tree and its parameters can be trained using gradient-descent. Our experimental results on regression, binary classification, and multi-class classification data sets indicate that our newly proposed model has better performance than traditional trees in terms of accuracy while inducing trees of comparable size.

Discriminative language modeling (DLM) is a feature-based approach that is used as an error-correcting step after hypothesis generation in automatic speech recognition (ASR). We formulate this both as a classification and a ranking problem and employ the perceptron, the margin infused relaxed algorithm (MIRA) and the support vector machine (SVM). To decrease training complexity, we try count-based thresholding for feature selection and data sampling from the list of hypotheses. On a Turkish morphology based feature set we examine the use of first and higher order$n$-grams and present an extensive analysis on the complexity and accuracy of the models with an emphasis on statistical significance. We find that we can save significantly from computation by feature selection and data sampling, without significant loss in accuracy. Using the MIRA or SVM does not lead to any further improvement over the perceptron but the use of ranking as opposed to classification leads to a 0.4% reduction in word error rate (WER) which is statistically significant.

In many bioinformatics applications, it is important to assess and compare the performances of algorithms trained from data, to be able to draw conclusions unaffected by chance and are therefore significant. Both the design of such experiments and the analysis of the resulting data using statistical tests should be done carefully for the results to carry significance. In this paper, we first review the performance measures used in classification, the basics of experiment design and statistical tests. We then give the results of our survey over 1,500 papers published in the last two years in three bioinformatics journals (including this one). Although the basics of experiment design are well understood, such as resampling instead of using a single training set and the use of different performance metrics instead of error, only 21 percent of the papers use any statistical test for comparison. In the third part, we analyze four different scenarios which we encounter frequently in the bioinformatics literature, discussing the proper statistical methodology as well as showing an example case study for each. With the supplementary software, we hope that the guidelines we discuss will play an important role in future studies.

Multiple kernel learning (MKL) uses a weighted combination of kernels where the weight of each kernel is optimized during training. However, MKL assigns the same weight to a kernel over the whole input space. Our main objective is the formulation of the localized multiple kernel learning (LMKL) framework that allows kernels to be combined with different weights in different regions of the input space by using a gating model. In this paper, we apply the LMKL framework to regression estimation and derive a learning algorithm for this extension. Canonical support vector regression may over fit unless the kernel parameters are selected appropriately; we see that even if provide more kernels than necessary, LMKL uses only as many as needed and does not overfit due to its inherent regularization.

We define the problem of optimizing the architecture of a multilayer perceptron (MLP) as a state space search and propose the MOST (Multiple Operators using Statistical Tests) framework that incrementally modifies the structure and checks for improvement using cross-validation. We consider five variants that implement forward/backward search, using single/multiple operators and searching depth-first/breadth-first. On 44 classification and 30 regression datasets, we exhaustively search for the optimal and evaluate the goodness based on: (1) Order, the accuracy with respect to the optimal and (2) Rank, the computational complexity. We check for the effect of two resampling methods (5 × 2, ten-fold cv), four statistical tests (5 × 2 cv t, ten-fold cv t, Wilcoxon, sign) and two corrections for multiple comparisons (Bonferroni, Holm). We also compare with Dynamic Node Creation (DNC) and Cascade Correlation (CC). Our results show that: (1) On most datasets, networks with few hidden units are optimal, (2) forward searching finds simpler architectures, (3) variants using single node additions (deletions) generally stop early and get stuck in simple (complex) networks, (4) choosing the best of multiple operators finds networks closer to the optimal, (5) MOST variants generally find simpler networks having lower or comparable error rates than DNC and CC.

Recently, instead of selecting a single kernel, multiple kernel learning (MKL) has been proposed which uses a convex combination of kernels, where the weight of each kernel is optimized during training. However, MKL assigns the same weight to a kernel over the whole input space. In this paper, we develop a localized multiple kernel learning (LMKL) algorithm using a gating model for selecting the appropriate kernel function locally. The localizing gating model and the kernel-based classifier are coupled and their optimization is done in a joint manner. Empirical results on ten benchmark and two bioinformatics data sets validate the applicability of our approach. LMKL achieves statistically similar accuracy results compared with MKL by storing fewer support vectors. LMKL can also combine multiple copies of the same kernel function localized in different parts. For example, LMKL with multiple linear kernels gives better accuracy results than using a single linear kernel on bioinformatics data sets.

Tao, et al have recently proposed the posterior probability support vector machine (PPSVM) which uses soft labels derived from estimated posterior probabilities to be more robust to noise and outliers. Tao, et al's model uses a window-based density estimator to calculate the posterior probabilities and is a binary classifier. We propose a neighbor-based density estimator and also extend the model to the multiclass case. Our bias-variance analysis shows that the decrease in error by PPSVM is due to a decrease in bias. On 20 benchmark data sets, we observe that PPSVM obtains accuracy results that are higher or comparable to those of canonical SVM using significantly fewer support vectors.

Given a data set and a number of supervised learning algorithms, we would like to find the algorithm with the smallest expected error. Existing pairwise tests allow a comparison of two algorithms only; range tests and ANOVA check whether multiple algorithms have the same expected error and cannot be used for finding the smallest. We propose a methodology, the MultiTest algorithm, whereby we order supervised learning algorithms taking into account 1) the result of pairwise statistical tests on expected error (what the data tells us), and 2) our prior preferences, e.g., due to complexity. We define the problem in graph-theoretic terms and propose an algorithm to find the "best" learning algorithm in terms of these two criteria, or in the more general case, order learning algorithms in terms of their "goodness." Simulation results using five classification algorithms on 30 data sets indicate the utility of the method. Our proposed method can be generalized to regression and other loss functions by using a suitable pairwise test.

We discuss and test empirically the effects of six dimensions along which existing decision tree induction algorithms differ. These are: Node type (univariate versus multivariate), branching factor (two or more), grouping of classes into two if the tree is binary, error (impurity) measure, and the methods for minimization to find the best split vector and threshold. We then propose a new decision tree induction method that we name linear discriminant trees (LDT) which uses the best combination of these criteria in terms of accuracy, simplicity and learning time. This tree induction method can be univariate or multivariate. The method has a supervised outer optimization layer for converting a K > 2-class problem into a sequence of two-class problems and each two-class problem is solved analytically using Fisher's Linear Discriminant Analysis (LDA). On twenty datasets from the UCI repository, we compare the linear discriminant trees with the univariate decision tree methods C4.5 and C5.0, multivariate decision tree methods CART, OC1, QUEST, neural trees and LMDT. Our proposed linear discriminant trees learn fast, are accurate, and the trees generated are small.

In local feature-based face recognition systems, the topographical locations of feature extractors directly affect the discriminative power of a recognizer. Better recognition accuracy can be achieved by the determination of the positions of salient image locations. Most of the facial feature selection algorithms in the literature work with two assumptions: one, that the importance of each feature is independent of the other features, and two, that the kernels should be located at fiducial points. Under these assumption, one can only get a sub-optimal solution. In this paper, we present a methodology that tries to overcome this problem by relaxing the two assumptions using a formalism of subset selection problem. We use a number of feature selection algorithms and a genetic algorithm. Comparative results on the FERET dataset confirm the viability of our approach.

Parallel pattern recognition requires great computational resources; it is NP-complete. From an engineering point of view it is desirable to achieve good performance with limited resources. For this purpose, we develop a serial model for visual pattern recognition based on the primate selective attention mechanism. The idea in selective attention is that not all parts of an image give us information. If we can attend only to the relevant parts, we can recognize the image more quickly and using less resources. We simulate the primitive, bottom-up attentive level of the human visual system with a saliency scheme and the more complex, top-down, temporally sequential associative level with observable Markov models. In between, there is a neural network that analyses image parts and generates posterior probabilities as observations to the Markov model. We test our model first on a handwritten numeral recognition problem and then apply it to a more complex face recognition problem. Our results indicate the promise of this approach in complicated vision applications.

For pt.I see ibid., p.645-61 (2002). Part I of this paper defines the class of constructive unsupervised on-line learning simplified adaptive resonance theory (SART) clustering networks. Proposed instances of class SART are the symmetric fuzzy ART (S-Fuzzy ART) and the Gaussian ART (GART) network. In Part II of our work, a third network belonging to class SART, termed fully self-organizing SART (FOSART), is presented and discussed. FOSART is a constructive, soft-to-hard competitive, topology-preserving, minimum-distance-to-means clustering algorithm capable of: 1) generating processing units and lateral connections on an example-driven basis and 2) removing processing units and lateral connections on a minibatch basis. FOSART is compared with Fuzzy ART, S-Fuzzy ART, GART and other well-known clustering techniques (e.g., neural gas and self-organizing map) in several unsupervised learning tasks, such as vector quantization, perceptual grouping and 3-D surface reconstruction. These experiments prove that when compared with other unsupervised learning networks, FOSART provides an interesting balance between easy user interaction, performance accuracy, efficiency, robustness, and flexibility.

For pt.I see ibid., p.645-61 (2002). Part I of this paper defines the class of constructive unsupervised on-line learning simplified adaptive resonance theory (SART) clustering networks. Proposed instances of class SART are the symmetric fuzzy ART (S-Fuzzy ART) and the Gaussian ART (GART) network. In Part II of our work, a third network belonging to class SART, termed fully self-organizing SART (FOSART), is presented and discussed. FOSART is a constructive, soft-to-hard competitive, topology-preserving, minimum-distance-to-means clustering algorithm capable of: 1) generating processing units and lateral connections on an example-driven basis and 2) removing processing units and lateral connections on a minibatch basis. FOSART is compared with Fuzzy ART, S-Fuzzy ART, GART and other well-known clustering techniques (e.g., neural gas and self-organizing map) in several unsupervised learning tasks, such as vector quantization, perceptual grouping and 3-D surface reconstruction. These experiments prove that when compared with other unsupervised learning networks, FOSART provides an interesting balance between easy user interaction, performance accuracy, efficiency, robustness, and flexibility.

Univariate decision trees at each decision node consider the value of only one feature leading to axis-aligned splits. In a linear multivariate decision tree, each decision node divides the input space into two with a hyperplane. In a nonlinear multivariate tree, a multilayer perceptron at each node divides the input space arbitrarily, at the expense of increased complexity and higher risk of overfitting. We propose omnivariate trees where the decision node may be univariate, linear, or nonlinear depending on the outcome of comparative statistical tests on accuracy thus matching automatically the complexity of the node with the subproblem defined by the data reaching that node. Such an architecture frees the designer from choosing the appropriate node type, doing model selection automatically at each node. Our simulation results indicate that such a decision tree induction method generalizes better than trees with the same types of nodes everywhere and induces small trees.

In this work pose-invariant face recognition is attempted using a two-stage approach. In the first stage, the orientation of the face is recognized, and in the second stage, the face is recognized among a subset of faces with the same orientation in the training set. We have generated our own database and tried several different techniques for pose invariant face recognition. We have used both linear techniques such as principal component analysis (PCA) and linear discriminant analysis (LDA) and unsupervised clustering techniques such as C-means and fuzzy C-means. The classification algorithms used with all of these techniques are nearest mean and k-nearest neighbor (k-NN). This work presents a comparison of these techniques on our database. With all techniques, it is observed that the recognition performance is enhanced when view information is incorporated as a preliminary step.

Dietterich (1998) reviews five statistical tests and proposes the 5 x 2 cv t test for determining whether there is a significant difference between the error rates of two classifiers. In our experiments, we noticed that the 5 x 2 cv t test result may vary depending on factors that should not affect the test, and we propose a variant, the combined 5 x 2 cv F test, that combines multiple statistics to get a more robust test. Simulation results show that this combined version of the test has lower type I error and higher power than 5 x 2 cv proper.

We investigate techniques to combine multiple representations of a handwritten digit to increase classification accuracy without significantly increasing system complexity or recognition time. We compare multiexpert and multistage combination techniques and discuss in detail in a comparative manner methods for combining multiple learners: voting, mixture of experts, stacking, boosting and cascading. In pen based handwritten character recognition, the input is the dynamic movement of the pentip over the pressure sensitive tablet. There is also the image formed as a result of this movement. On a real world database, we notice that the two multi layer perceptron (MLP) neural network based classifiers using these representations separately make errors on different patterns, implying that a suitable combination of the two would lead to higher accuracy. Thus we implement and compare voting, mixture of experts, stacking and cascading. Combined classifiers have an error percentage less than individual ones. The final combined system of two MLPs has less complexity and memory requirement than a single k nearest neighbor using one of the representations.

Quantization errors are generally hidden by performing a dithering operation on the image. A common method is to utilize error diffusion. However, this method is prone to error accumulation, resulting in color impulses and streaks. This paper presents an improvement on error diffusion dithering through a fuzzy error diffusion algorithm. In this method, the amount of error to be diffused is determined by considering the relative location of the pixel not only to the closest codebook vector, but also to the color cluster it belongs to. The goal is to hide the quantization errors by error diffusion, while preventing the excess accumulation of errors. This is achieved through an attraction-repulsion schema according to a fuzzy membership function. We have implemented the fuzzy error diffusion algorithm for color images and achieved drastic improvements, resulting in superior quality dithered images and significantly lower mean squared error values.

A structure composed of local linear perceptrons for approximating global class discriminants is investigated. Such local linear models may be combined in a cooperative or competitive way. In the cooperative model, a weighted sum of the outputs of the local perceptrons is computed where the weight is a function of the distance between the input and the position of the local perceptron. In the competitive model, the cost function dictates a mixture model where only one of the local perceptrons give output. Learning of the local models' positions and the linear mappings they implement are coupled and both supervised. We show that this is preferable to the uncoupled case where the positions are trained in an unsupervised manner before the separate, supervised training of mappings. We use goodness criteria based on the cross-entropy and give learning equations for both the cooperative and competitive cases. The coupled and uncoupled versions of cooperative and competitive approaches are compared among themselves and with multilayer perceptrons of sigmoidal hidden units and radial basis functions (RBFs) of Gaussian units on the application of recognition of handwritten digits. The criteria of comparison are the generalization accuracy, learning time, and the number of free parameters. We conclude that even on such a high-dimensional problem, such local models are promising. They generalize much better than RBF's and use much less memory. When compared with multilayer perceptrons, we note that local models learn much faster and generalize as well and sometimes better with comparable number of parameters.

Learning when limited to modification of some parameters has a limited scope; capability to modify the system structure is also needed to get a wider range of the learnable. In the case of artificial neural networks, learning by iterative adjustment of synaptic weights can only succeed if the network designer predefines an appropriate network structure, i.e. the number of hidden layers, units, and the size and shape of their receptive and projective fields. This paper advocates the view that the network structure should not, as is usually done, be determined by trial-and-error but should be computed by the learning algorithm. Incremental learning algorithms can modify the network structure by addition and/or removal of units and/or links. A survey of current connectionist literature is given on this line of thought. “Grow and Learn” (GAL) is a new algorithm that learns an association at one shot due to its being incremental and using a local representation. During the so-called “sleep” phase, units that were previously stored but which are no longer necessary due to recent modifications are removed to minimize network complexity. The incrementally constructed network can later be finetuned off-line to improve performance. Another method proposed that greatly increases recognition accuracy is to train a number of networks and vote over their responses. The algorithm and its variants were tested on recognition of handwritten numerals and seem promising especially in terms of learning speed. This makes the algorithm attractive for on-line learning tasks, e.g. in robotics. The biological plausibility of incremental learning is also discussed briefly.

Proposes to train a number of neural networks independently, either with different learning algorithms or with the same algorithm but with different sets of parameters and then take a vote over them in the form of a weighted majority. All the nets learn essentially the same task but converge to different solutions due to different learning parameter values, e.g. net structure. The voting schemes investigated are static vs. dynamic, where in dynamic voting scheme, the network complexity is also taken into account. The system assigns a weight of 'confidence' to each participating net proportional to the net's success and complexity. An empirical work shows that having multiple nets significantly improves generalization, i.e., higher success is achieved in classifying previously unseen data. This framework is not limited to neural nets but can be applied to learning systems in general.>