Local Continuity Meta Criterion (LCMC)¶
Description¶
The Local Continuity Meta Criterion (LCMC) evaluates the quality of dimensionality reduction methods by measuring how well local neighborhood structures are preserved between the original high-dimensional space and the low-dimensional embedding. For each data point, it compares the set of its K nearest neighbors before and after projection. The metric quantifies the overlap and adjusts for chance-level matches. LCMC is particularly useful for identifying how well local geometry is maintained, complementing other metrics like Trustworthiness and Continuity.
Formulas¶
Let :
-
\(N\) : number of data points
-
\(K\) : neighborhood size
-
\(\mathcal{N}^K(i)\) : set of the \(K\) nearest neighbors of point \(i\) in the original space
-
\(\nu^K(i)\) : set of the \(K\) nearest neighbors of point \(i\) in the projected space
Then the LCMC is defined as :
This means:
-
The first term is the average number of overlapping neighbors (shared between both spaces), normalized.
-
The second term \(\frac{K}{N-1}\) is the expected overlap by chance.
This adjustment means that:
-
LCMC = 1 implies perfect local continuity (all true neighbors preserved),
-
LCMC ≈ 0 implies performance similar to random neighbor selection,
-
LCMC < 0 suggests worse-than-random behavior.
Sources¶
“Applying Deep Learning algorithm to perform lung cells annotation”, A. Collin, 2020
Jarkko VENNA et Samuel KASKI. « Local multidimensional scaling ». In : Neural Networks (2006).