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Multi-step Training of a Generalized Linear Classifier

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Abstract

We propose a multi-step training method for designing generalized linear classifiers. First, an initial multi-class linear classifier is found through regression. Then validation error is minimized by pruning of unnecessary inputs. Simultaneously, desired outputs are improved via a method similar to the Ho-Kashyap rule. Next, the output discriminants are scaled to be net functions of sigmoidal output units in a generalized linear classifier. This classifier is trained via Newton’s algorithm. Performance gains are demonstrated at each step. Using widely available datasets, the final network’s tenfold testing error is shown to be less than that of several other linear and generalized linear classifiers reported in the literature.

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Authors and Affiliations

  1. Department of Electrical Engineering, The University of Texas at Arlington, Arlington, TX, 76019, USA

    Kanishka Tyagi & Michael Manry

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  1. Kanishka Tyagi
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  2. Michael Manry
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Correspondence to Kanishka Tyagi.

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A Appendix

A Appendix

1.1 A.1 Datasets

In order to evaluate the performance of all the improvements and our proposed algorithm, we used many publicly available datasets. Table 5 tabulate the specifications for these datasets. It should be noted here that all the datasets that we have used in our experiments have balanced classes.

Table 5 Specification of datasets
Full size table

1.1.1 A.1.1 Gongtrn Dataset

The raw data consists of images from hand printed numerals [57] collected from 3000 people by the Internal Revenue Service. We randomly chose 300 characters from each class to generate 3000 character training data. Images are 32 by 24 binary matrices. An image scaling algorithm is used to remove size variation in characters. The feature set contains 16 elements. The 10 classes correspond to 10 Arabic numerals.

1.1.2 A.1.2 Comf18 Dataset

The training data file is generated from segmented images [58]. Each segmented region is separately histogram equalized to 20 levels. Then the joint probability density of pairs of pixels separated by a given distance and a given direction is estimated. We use 0, 90, 180, 270 degrees for the directions and 1, 3, and 5 pixels for the separations. The density estimates are computed for each classification window. For each separation, the co-occurrences for for the four directions are folded together to form a triangular matrix. From each of the resulting three matrices, six features are computed: angular second moment, contrast, entropy, correlation, and the sums of the main diagonal and the first off diagonal. This results in 18 features for each classification window.

1.1.3 A.1.3 MNIST Dataset

The digits data used in this book is taken from the MNIST data set [59], which itself was constructed by modifying a subset of the much larger dataset produced by NIST (the National Institute of Standards and Technology). It comprises a training set of 60,000 examples and a test set of 10,000 examples. The original NIST data had binary (black or white) pixels. To create MNIST,these images were size normalized to fit in a 20 20 pixel box while preserving their aspect ratio. As a consequence of the anti-aliasing used to change the resolution of the images, the resulting MNIST digits are grey scale. These images were then centered in a 28 28 box. This dataset is a classic within the machine learning community and has been extensively studied.

1.1.4 A.1.4 Google Street View Dataset

The Google street view housing numbers (SVHN) [60] is a real-world image dataset for developing machine learning and object recognition algorithms with minimal requirement on data preprocessing and formatting. It can be seen as similar in flavor to MNIST (e.g., the images are of small cropped digits), but incorporates an order of magnitude more labeled data (over 600,000 digit images) and comes from a significantly harder, unsolved, real world problem (recognizing digits and numbers in natural scene images). SVHN is obtained from house numbers in Google Street View images.

1.1.5 A.1.5 CIFAR Dataset

The CIFAR-10 dataset [61] consists of 60,000 \(32\times 32\) colour images in 10 classes, with 6000 images per class. There are 50,000 training images and 10,000 test images. The dataset is divided into five training batches and one test batch, each with 10,000 images. The test batch contains exactly 1000 randomly-selected images from each class. The training batches contain the remaining images in random order, but some training batches may contain more images from one class than another. Between them, the training batches contain exactly 5000 images from each class.

1.1.6 A.1.6 COVER

This dataset [62] is contains forest cover type for a given observation (\(30\times 30\) meter cell) that was determined from US Forest Service (USFS) Region 2 Resource Information System (RIS) data. Independent variables were derived from data originally obtained from US Geological Survey (USGS) and USFS data. Data is in raw form (not scaled) and contains binary (0 or 1) columns of data for qualitative independent variables (wilderness areas and soil types).

1.1.7 A.1.7 RCV1

Reuters Corpus Volume I (RCV1) [63] is an archive of over 800,000 manually categorized newswire stories made available by Reuters, Ltd. for research purposes. The dataset is extensively described in [1].

1.1.8 A.1.8 NEWS-20

The 20 Newsgroups dataset [64] is a collection of approximately 20,000 newsgroup documents, partitioned (nearly) evenly across 20 different newsgroups. The 20 newsgroups collection has become a popular data set for experiments in text applications of machine learning techniques, such as text classification and text clustering.

1.1.9 A.1.9 Breast cancer

The breast cancer dataset [65] is a collection of 989 features that are reduced in dimension using principal component analysis to 42 features.

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Tyagi, K., Manry, M. Multi-step Training of a Generalized Linear Classifier. Neural Process Lett 50, 1341–1360 (2019). https://doi.org/10.1007/s11063-018-9915-4

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