The successive subspace learning (SSL) principle was developed and used to design an interpretable learning model, known as the PixelHop method,for image classification in our prior work. Here, we propose an improved PixelHop method and call it PixelHop++. First, to make the PixelHop model size smaller, we decouple a joint spatial-spectral input tensor to multiple spatial tensors (one for each spectral component) under the spatial-spectral separability assumption and perform the Saab transform in a channel-wise manner, called the channel-wise (c/w) Saab transform.Second, by performing this operation from one hop to another successively, we construct a channel-decomposed feature tree whose leaf nodes contain features of one dimension (1D). Third, these 1D features are ranked according to their cross-entropy values, which allows us to select a subset of discriminant features for image classification. In PixelHop++, one can control the learning model size of fine-granularity,offering a flexible tradeoff between the model size and the classification performance. We demonstrate the flexibility of PixelHop++ on MNIST, Fashion MNIST, and CIFAR-10 three datasets.

A new machine learning methodology, called successive subspace learning (SSL), is introduced in this work. SSL contains four key ingredients: 1) successive near-to-far neighborhood expansion; 2) unsupervised dimension reduction via subspace approximation; 3) supervised dimension reduction via label-assisted regression (LAG); and 4) feature concatenation and decision making. An image-based object classification method, called PixelHop, is proposed to illustrate the SSL design. It is shown by experimental results that the PixelHop method outperforms the classic CNN model of similar model complexity in three benchmarking datasets (MNIST, Fashion MNIST and CIFAR-10). Although SSL and deep learning (DL) have some high-level concept in common, they are fundamentally different in model formulation, the training process and training complexity. Extensive discussion on the comparison of SSL and DL is made to provide further insights into the potential of SSL.