# summary

Dimensionality Reduction by Learning an Invariant Mapping

## 1. Intention

The drawbacks of most existing technique:

1 Most of them depend on a meaningful and computable distance metric in input space. (eg. LLE, Isomap relies on computable distance)

2 They do not compute a “function” that can accurately map new input samples whose relationship to the training data is unknown.

To overcome these drawbacks, this paper introduces a technique called DrLIM. The learning relies solely on neighborhood relationships and does not require any distance measure in the input space.

## 2. Mathematical Model

Input: A set of vectors [math] I=\{x_1,x_2,......,x_p\} [/math], where [math] x_i\in \mathbb{R}^D, \forall i=1,2,3......,n. [/math] Output: A parametric function [math]G_W:\mathbb{R}^D \rightarrow \mathbb{R}^d [/math] with [math] d\lt \lt D [/math]

The optimization problem of BoostMetric is similar to the large margin nearest neighbor algorithm (LMNN [4]). In the preprocessing step, the labeled training samples are required to be transformed into "triplets" (a _{i}, a _{j}, a _{k}), where a _{i} and a _{j} are in the same class, but a _{i} and a _{k} are in different classes. Let us denote dist _{i,j} and dist_{i,k} as the distance between a_{i} and a_{j} and the distance between a_{i} and a_{k} separately. The goal is to maximize the difference between these two distances.

Here the distance is Mahalanobis matrix represented as follows:

[math]dist_{ij}^{2}=\left \| L^Ta_i-L^Ta_j \right \|_2^2=(a_i-a_j)^TLL^T(a_i-a_j)=(a_i-a_j)^TX(a_i-a_j).[/math]