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View incremental decremental multi-view discriminant analysis

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Abstract

The majority of multi-view methods in the literature are batch methods that need all data at the beginning of a training session. However, these methods are not scalable and need to retrain if new data are added or removed from the existing dataset. Incremental methods that support the addition of data samples have been developed, but these methods do not support the addition of views. To address this issue, we present view incremental decremental multi-view discriminant analysis (VIDMvDA) that updates a learned model without retraining when new views are added or existing views are deleted. VIDMvDA is presented in two forms: incremental learning and decremental unlearning. It provides closed-form solutions to update the within-class and the between-class scatter. We have measured the performance of our method against multi-view batch methods on criteria such as discriminability, order independence, classification accuracy, training time, and memory. We prove that using significantly less training time and memory, VIDMvDA constructs a similar discriminant subspace and has the same or better classification accuracy than the batch methods.

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Data Availability

The datasets used in this study are publicly available at the following links:

– Handwritten Digits Dataset

– Caltech-7 Dataset

– AwA Dataset

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Authors and Affiliations

  1. Department of CSE, IIT Guwahati, Assam, India

    Saroj S. Shivagunde & V. Vijaya Saradhi

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  1. Saroj S. Shivagunde
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Correspondence to Saroj S. Shivagunde.

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Appendices

Appendix A: Deriving the within-class scatter equation

$$ \begin{array}{@{}rcl@{}} \textbf{S}_{W}^{\prime} &=& \sum\limits_{i=1}^{c} \sum\limits_{j=1}^{v} \sum\limits_{k=1}^{n_{ij}} \left( \textbf{y}_{ijk} - \boldsymbol{\mu}_{i}^{\prime} \right) \left( \textbf{y}_{ijk} - \boldsymbol{\mu}_{i}^{\prime} \right)^{T}\\ &&+ \sum\limits_{i=1}^{c} \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i}^{\prime} \right) \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i}^{\prime} \right)^{T}\\ &=& \sum\limits_{i=1}^{c} \sum\limits_{j=1}^{v} \sum\limits_{k=1}^{n_{ij}} \left( \textbf{y}_{ijk} - \boldsymbol{\mu}_{i} - \frac{l_{i} (\textbf{m}_{i}-\boldsymbol{\mu}_{i} )}{n_{i} + l_{i}} \right) \left( \textbf{y}_{ijk} - \boldsymbol{\mu}_{i} - \frac{l_{i} (\textbf{m}_{i}-\boldsymbol{\mu}_{i} )}{n_{i} + l_{i}} \right)^{T}\\ && + \sum\limits_{i=1}^{c} \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i} - \frac{l_{i} (\textbf{m}_{i} - \boldsymbol{\mu}_{i} )}{n_{i} + l_{i}} \right) \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i} - \frac{l_{i} (\textbf{m}_{i}-\boldsymbol{\mu}_{i} )}{n_{i} + l_{i}} \right)^{T} \\ &=& \textbf{S}_{W} + \sum\limits_{i=1}^{c} \frac{1}{(n_{i} + l_{i})^{2}} \left[\vphantom{\sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}}} n_{i} (l_{i})^{2} \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right) \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right)^{T}\right.\\ &&\left.+ {n_{i}^{2}} \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i} \right) \left( \textbf{y}_{ijk}^{\prime} - \boldsymbol{\mu}_{i} \right)^{T}+ (2 n_{i} l_{i}\right.\\ &&\left.+ {l_{i}^{2}}) \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \left( \textbf{y}_{ijk}^{\prime} - \textbf{m}_{i} \right) \left( \textbf{y}_{ijk}^{\prime} - \textbf{m}_{i} \right)^{T} \right] \\ &=& \textbf{S}_{W} + \sum\limits_{i=1}^{c} \left[\vphantom{\sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}}} \frac{l_{i} n_{i}}{n_{i} + l_{i}} \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right) \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right)^{T}\right.\\ &&\left.+ \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \left( \textbf{y}_{ijk}^{\prime} - \textbf{m}_{i} \right) \left( \textbf{y}_{ijk}^{\prime} - \textbf{m}_{i} \right)^{T} \right] \\ &=& \textbf{S}_{W} + \sum\limits_{i=1}^{c} \left[\vphantom{\sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}}} \frac{l_{i} n_{i}}{n_{i} + l_{i}} \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right) \left( \textbf{m}_{i} -\boldsymbol{\mu}_{i} \right)^{T}\right.\\ &&\left.+ \sum\limits_{j \in \mathscr{A}} \sum\limits_{k=1}^{l_{ij}} \textbf{y}_{ijk}^{\prime} {\textbf{y}_{ijk}^{\prime}}^{T} - l_{i} \textbf{m}_{i} \textbf{m}_{i}^{T} \right] \\ &=& \sum\limits_{j=1}^{v} \sum\limits_{r=1}^{v} \textbf{W}_{j}^{T} \left[ \textbf{S}_{jr} + \sum\limits_{i=1}^{c} \frac{l_{i} n_{ij} n_{ir}}{n_{i} (n_{i} + l_{i})} {\boldsymbol{\mu}_{ij}}^{(\textbf{x})} {\boldsymbol{\mu}_{ir}}^{(\textbf{x})^{T}} \right] \textbf{W}_{r}\\ &&+ \sum\limits_{j \in \mathscr{A}} \textbf{W}_{j}^{T} \left[ {\sum}_{i=1}^{c} \sum\limits_{k=1}^{l_{ij}} \textbf{x}_{ijk}^{\prime} {\textbf{x}_{ijk}^{\prime}}^{T} \right] \textbf{W}_{j} \\ && - \sum\limits_{j \in \mathscr{A}} \sum\limits_{r \in \mathscr{A}} \textbf{W}_{j}^{T} \left[ \sum\limits_{i=1}^{c} \frac{l_{ij} l_{ir}}{l_{i}} \textbf{m}_{ij}^{(x)} \textbf{m}_{ir}^{(x)^{T}} \right] \textbf{W}_{r} \\ &=& \sum\limits_{j=1}^{v} \sum\limits_{r=1}^{v} \textbf{W}_{j}^{T} \textbf{S}_{jr}^{\prime} \textbf{W}_{r} = \textbf{W}^{T} \textbf{S}^{\prime} \textbf{W} \end{array} $$

Where,

$$ \textbf{S}_{jr}^{\prime} = \left\{\begin{array}{ll} - \sum\limits_{i=1}^{c} \frac{l_{ij} l_{ir}}{l_{i}} {\textbf{m}_{ij}}^{(\textbf{x})} {\textbf{m}_{ir}}^{(\textbf{x})^{T}} & \text{case 1} \\[0.4cm] \sum\limits_{i=1}^{c} \bigg[ \sum\limits_{k=1}^{l_{ij}} \textbf{x}_{ijk}^{\prime} {\textbf{x}^{\prime}_{ijk}}^{T} - \frac{l_{ij} l_{ij}}{l_{i}} {\textbf{m}_{ij}}^{(\textbf{x})} {\textbf{m}_{ij}}^{(\textbf{x})^{T}} \bigg] & \text{case 2} \\[0.4cm] \textbf{S}_{jr} + \sum\limits_{i=1}^{c} \frac{n_{i}^{\prime} - n_{i}}{n_{i} n_{i}^{\prime}} n_{ij} n_{ir} {\boldsymbol{\mu}_{ij}}^{(\textbf{x})} {\boldsymbol{\mu}_{ir}}^{(\textbf{x})^{T}} & \text{case 3} \end{array}\right. $$

Appendix B: Deriving the between-class scatter equation

$$ \textbf{S}_{B}^{\prime} = \sum\limits_{i=1}^{c} n_{i}^{\prime} (\boldsymbol{\mu}_{i}^{\prime} - \boldsymbol{\mu}^{\prime}) (\boldsymbol{\mu}_{i}^{\prime} - \boldsymbol{\mu}^{\prime})^{T} $$
$$ \begin{array}{@{}rcl@{}} \textbf{S}_{B}^{\prime} &=&\sum\limits_{i=1}^{c} n_{i}^{\prime} \boldsymbol{\mu}_{i}^{\prime} {\boldsymbol{\mu}_{i}^{\prime}}^{T} - \sum\limits_{i=1}^{c} n_{i}^{\prime} \boldsymbol{\mu}_{i}^{\prime} {\boldsymbol{\mu}^{\prime}}^{T} - \sum\limits_{i=1}^{c} n_{i}^{\prime} \boldsymbol{\mu}^{\prime} {\boldsymbol{\mu}_{i}^{\prime}}^{T} + \sum\limits_{i=1}^{c} n_{i}^{\prime} \boldsymbol{\mu}^{\prime} {\boldsymbol{\mu}^{\prime}}^{T}\\ &=&\sum\limits_{i=1}^{c} n_{i}^{\prime} \boldsymbol{\mu}_{i}^{\prime} {\boldsymbol{\mu}_{i}^{\prime}}^{T} - n^{\prime} \boldsymbol{\mu}^{\prime} {\boldsymbol{\mu}^{\prime}}^{T}\\ &=& \sum\limits_{i=1}^{c} n_{i}^{\prime} \left( \sum\limits_{j=1}^{v^{\prime}} \frac{n_{ij}^{\prime} }{n_{i}^{\prime}} \textbf{W}_{j}^{T} \boldsymbol{\mu}_{ij}^{\prime} \right) \left( \sum\limits_{r=1}^{v^{\prime}} \frac{n_{ir}^{\prime}}{n_{i}^{\prime}} \textbf{W}_{r}^{T} \boldsymbol{\mu}_{ir}^{\prime} \right)^{T} \\ && - n^{\prime} \left( \sum\limits_{j=1}^{v^{\prime}} \sum\limits_{i=1}^{c} \frac{n_{ij}^{\prime}}{n^{\prime}} \textbf{W}_{j}^{T} \boldsymbol{\mu}_{ij}^{\prime} \right) \left( \sum\limits_{r=1}^{v^{\prime}} \sum\limits_{i=1}^{c} \frac{n_{ij}^{\prime}}{n^{\prime}} \textbf{W}_{r}^{T} \boldsymbol{\mu}_{ir}^{\prime} \right)^{T}\\ &=&\sum\limits_{j=1}^{v^{\prime}} \sum\limits_{r=1}^{v^{\prime}} \textbf{W}_{j}^{T} \textbf{D}_{jr}^{\prime} \textbf{W}_{r} \\ \end{array} $$

Where,

$$ \begin{array}{@{}rcl@{}} \textbf{D}_{jr}^{\prime} &=& \sum\limits_{i=1}^{c} \frac{n_{ij}^{\prime} n_{ir}^{\prime}}{n_{i}^{\prime}} {\boldsymbol{\mu}_{ij}^{\prime}}^{(x)} {{\boldsymbol{\mu}_{ir}^{\prime}}^{(x)}}^{T}\\ &&- \frac{1}{n^{\prime}} \left( \sum\limits_{i=1}^{c} n_{ij}^{\prime} {\boldsymbol{\mu}_{ij}^{\prime}}^{(x)} \right) \left( \sum\limits_{i=1}^{c} n_{ir}^{\prime} {{\boldsymbol{\mu}_{ir}^{\prime}}^{(x)}}^{T} \right) \end{array} $$

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Shivagunde, S.S., Saradhi, V.V. View incremental decremental multi-view discriminant analysis. Appl Intell 53, 13593–13607 (2023). https://doi.org/10.1007/s10489-022-04168-x

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