Abstract |
Many signal processing applications rely on multidimensional signals, which require demanding acquisition and analysis procedures. Tensors, which are multidimensional arrays indexed by multiple variables, are a natural way to represent these signals. Despite
the complexities involved in handling and manipulating tensors, tools for tensor analysis can effectively overcome the limitations of traditional processing models. This thesis
presents innovative tensor learning techniques that tackle the challenges associated with
high-dimensional signal acquisition and analysis.
During the data acquisition process, a common issue is the corruption or loss of a
significant number of measurements due to communication failures. Additionally, the
available measurements are often quantized to a specific number of bits for transmission
purposes. To enable further analysis, a quantization model for high-dimensional data is
proposed, along with formal methods for recovering a tensor from partially quantized and
potentially corrupted measurements. The study also investigates the relationship between
quantization and sampling, as well as the identification of possible anomalies or outliers
in higher-order signals. Experiments conducted on satellite-derived observations indicate
that it is more effective to consider the discrete nature of the quantized measurements
rather than treating them as real values.
The compression process is another critical step in signal acquisition that aims to reduce the required number of bits for representation without significant loss. To this end,
we introduce end-to-end compression algorithms for high-dimensional data that include
quantization and coding to bitstreams, allowing them to be directly applied in real-world
scenarios. In these schemes, we populate the relevant representation subspaces and utilize tensor decomposition techniques within a machine-learning framework to exploit the
data characteristics of available training samples. Subsequently, each new sample can be
mapped to the learned decompositions, necessitating only the transmission of associated
coefficients. Consequently, high compression ratios can be achieved, and the correlations
among all variables are simultaneously removed. While these algorithms are evaluated on
third- and fourth-order remote sensing multispectral image sequences, they can handle
arbitrary-high-dimensional data, providing a mathematically concrete solution for encoding multiple sources of observations simultaneously.
Multidimensional measurements have also significant implications for their analysis,
including classification. Therefore, this dissertation proposes a general machine-learning
approach for supervised classification based on tensor decomposition. The problem is for
mulated as a tensor completion task, where the tensor of scores for various classes and all
possible samples, is completed. By integrating classification loss functions with tensor decomposition techniques, the proposed method is effectively utilized in several real-world
classification tasks.
In conclusion, the proposed schemes effectively tackle numerous challenges associated with high-dimensional signal acquisition and processing. However, the main contribution of this thesis lies in the integration of tensor models with machine learning techniques. By leveraging the advantages of both tensor decomposition and machine learning,
we can concurrently exploit their benefits and provide a novel framework for multidimensional signal processing that overcomes the constraints of conventional models.
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