  In this work, we present tensor-based linear and nonlinear models for
hyperspectral data classification and analysis. By exploiting principles of
tensor algebra, we introduce new classification architectures, the weight
parameters of which satisfies the {\it rank}-1 canonical decomposition
property. Then, we introduce learning algorithms to train both the linear and
the non-linear classifier in a way to i) to minimize the error over the
training samples and ii) the weight coefficients satisfies the {\it rank}-1
canonical decomposition property. The advantages of the proposed classification
model is that i) it reduces the number of parameters required and thus reduces
the respective number of training samples required to properly train the model,
ii) it provides a physical interpretation regarding the model coefficients on
the classification output and iii) it retains the spatial and spectral
coherency of the input samples. To address issues related with linear
classification, characterizing by low capacity, since it can produce rules that
are linear in the input space, we introduce non-linear classification models
based on a modification of a feedforward neural network. We call the proposed
architecture {\it rank}-1 Feedfoward Neural Network (FNN), since their weights
satisfy the {\it rank}-1 caconical decomposition property. Appropriate learning
algorithms are also proposed to train the network. Experimental results and
comparisons with state of the art classification methods, either linear (e.g.,
SVM) and non-linear (e.g., deep learning) indicates the outperformance of the
proposed scheme, especially in cases where a small number of training samples
are available. Furthermore, the proposed tensor-based classfiers are evaluated
against their capabilities in dimensionality reduction.
