Chebyshev distance

Chebyshev distance is named after Pafnuty Chebyshev

Chebyshev distance is a metric used in mathematics to measure the distance between two points in a coordinate space. It is defined as the greatest of their differences along any coordinate dimension. This metric is particularly useful in situations where movement is restricted to grid-like paths, such as in chess or urban grid layouts.

Example

In chess, the minimum number of moves needed by a king to go from one square to another equals the Chebyshev distance between the centers of the squares. For instance, moving from f6 to e2 requires 4 moves, which corresponds to the Chebyshev distance of 4.

Understanding Chebyshev distance helps in analyzing movements and optimizing paths in grid-like environments, such as chess or urban planning.

Related concepts

Euclidean geometry

Euclidean distance measures absolute position in space

cosine similarity works better than Euclidean distance in high dimensions

Cosine similarity measures orientation, not magnitude, making it more robust to irrelevant dimensions in high-dimensional spaces

Distance transform

Manhattan distance formula: |x1 - x2| + |y1 - y2|

random projection to O(log n/ε²) dimensions preserves pairwise distances within 1±ε

Random projection reduces dimensionality while preserving pairwise distances within ε² due to the Johnson-Lindenstrauss lemma

the Johnson-Lindenstrauss lemma says

Random projection reduces dimensionality while approximately preserving pairwise distances

non-convex loss landscapes are hard: many local minima and saddle points

Non-convex loss landscapes are hard due to many local minima and saddle points

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