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tinybig.interdependence

This module provides the "data interdependence functions" that can be used to build the RPN model within the tinyBIG toolkit.

Data Interdependence Function

The data interdependence functions \(\xi\) capture the intricate interdependence relationships among data instances and attributes. These functions will extract nuanced information from the input data batch, operating both prior to and following the data projection facilitated by function \(\kappa\).

Formally, given an input data batch \(\mathbf{X} \in {R}^{b \times m}\) (with \(b\) instances and each instance with \(m\) attributes), the attribute and instance data interdependence functions are defined as:

\[
    \begin{equation}
    \xi_a: {R}^{b \times m} \to {R}^{m \times m'} \text{, and }
    \xi_i: {R}^{b \times m} \to {R}^{b \times b'},
    \end{equation}
\]

where \(m'\) and \(b'\) denote the output dimensions of their respective interdependence functions, respectively.

Data Interdependent Transformation Function

To elucidate the mechanisms of attribute and instance interdependence functions in defining the data interdependence transformation function \(\kappa_{\xi}\), we shall consider a multi-instance input data batch \(\mathbf{X} \in {R}^{b \times m}\) as an exemplar. Here, \(b\) and \(m\) denote the number of instances and attributes, respectively. Given this input data batch \(\mathbf{X}\), we can formulate the data interdependence transformation function \(\kappa_{\xi}\) as follows:

\[
    \begin{equation}
    \kappa_{\xi}(\mathbf{X}) = \mathbf{A}^\top_{\xi_i} \kappa(\mathbf{X} \mathbf{A}_{\xi_a}) \in {R}^{b' \times D}.
    \end{equation}
\]

These attribute and instance interdependence matrices \(\mathbf{A}_{\xi_a} \in {R}^{m \times m'}\) and \(\mathbf{A}_{\xi_i} \in {R}^{b \times b'}\) are computed with the corresponding interdependence functions defined above, i.e.,

\[
    \begin{equation}
    \mathbf{A}_{\xi_a} = \xi_a(\mathbf{X}) \in {R}^{m \times m'} \text{, and } \mathbf{A}_{\xi_i} = \xi_i(\mathbf{X}) \in {R}^{b \times b'}.
    \end{equation}
\]

The dimension of the target transformation space, denoted as \(D\), is determined by the codomain dimension \(m'\) of the attribute interdependence function. In most cases, the domain and codomain dimensions of the attribute and instance dependence functions analyzed in this paper are identical, i.e., \(m' = m\) and \(b' = b\).

Classes in this Module

This module contains the following categories of compression functions:

  • Basic interdependence functions
  • Geometric interdependence functions (based on the cuboid, cylinder and sphere patch shapes)
  • Topological interdependence functions (based on graph and chain structures)
  • Kernel based interdependence functions
  • Parameterized interdependence functions
  • Parameterized bilinear interdependence functions
  • Parameterized RPN based interdependence functions
  • Hybrid interdependence functions

Organization of this Module

Interdependence Function

Basic Interdependence Functions

Statistical Kernel based Interdependence Functions

Numerical Kernel based Interdependence Functions

Parameterized Interdependence Functions

Parameterized Bilinear Interdependence Functions

Topological Structural Interdependence Functions

Geometric Structural Interdependence Functions

Parameterized RPN based Interdependence Function

Hybrid Interdependence Function