Curse of Dimensionality
A phenomenon identified by mathematician Richard Bellman in 1957, describing how the sample size and computational cost required to solve a problem grow exponentially as the number of dimensions increases, rendering traditional grid-based or kernel-based density estimation methods nearly ineffective in high-dimensional spaces.
Timeline (last 90 days)
Vectorizing a tensor destroys the multidimensional structure of the data and leads to an explosion in the number of parameters; if a p1×p2×…×pK tensor is directly flattened into a vector, its dimension is the product of the dimensions of all modes, and the number of parameters grows exponentially with the number of modes.
When the intrinsic dimension d is large, exponentially more samples are needed to achieve the same accuracy.
Existing convergence rate analyses typically depend on the ambient dimension, causing theoretically predicted convergence speeds to nearly break down in high dimensions, i.e., the curse of dimensionality.
In nonparametric density estimation, the number of samples required to achieve ε accuracy grows at a rate of ε^(-D) with the ambient dimension D.
In high-dimensional spaces, distances between data points tend to become uniform, causing distance-based algorithms to perform significantly worse; this is called the curse of dimensionality
All Facts (5)
Vectorizing a tensor destroys the multidimensional structure of the data and leads to an explosion in the number of parameters; if a p1×p2×…×pK tensor is directly flattened into a vector, its dimension is the product of the dimensions of all modes, and the number of parameters grows exponentially with the number of modes.
50%UnverifiedExisting convergence rate analyses typically depend on the ambient dimension, causing theoretically predicted convergence speeds to nearly break down in high dimensions, i.e., the curse of dimensionality.
50%UnverifiedIn nonparametric density estimation, the number of samples required to achieve ε accuracy grows at a rate of ε^(-D) with the ambient dimension D.
50%UnverifiedIn high-dimensional spaces, distances between data points tend to become uniform, causing distance-based algorithms to perform significantly worse; this is called the curse of dimensionality
50%UnverifiedWhen the intrinsic dimension d is large, exponentially more samples are needed to achieve the same accuracy.
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