# Pointed subspace approach to incomplete data

## TL;DR

This paper introduces a novel representation of incomplete data as pointed affine subspaces, enabling affine transformations and embedding into vector space for improved classification handling of missing data.

## Contribution

It generalizes the traditional missing data representation by using pointed affine subspaces and provides a method to embed these into vector space while preserving scalar products.

## Key findings

- Allows affine transformations of incomplete data
- Enables embedding into vector space for classification
- Preserves scalar products in the embedding

## Abstract

Incomplete data are often represented as vectors with filled missing attributes joined with flag vectors indicating missing components. In this paper we generalize this approach and represent incomplete data as pointed affine subspaces. This allows to perform various affine transformations of data, as whitening or dimensionality reduction. We embed such generalized missing data into a vector space by mapping pointed affine subspace (generalized missing data point) to a vector containing imputed values joined with a corresponding projection matrix. Such an operation preserves the scalar product of the embedding defined for flag vectors and allows to input transformed incomplete data to typical classification methods.