Natarajan Dimension
A combinatorial measure of the complexity of a hypothesis class, introduced by Natarajan in 1989. It is commonly used to analyze the sample complexity of multiclass classification and agnostic learning, and is based on the number of projection patterns that the hypothesis class induces on sample points.
Timeline (last 90 days)
In the agnostic learning setting, the gap between the sample complexity of every ERM and the known lower bound is at most the ratio of the DS dimension to the Natarajan dimension multiplied by a constant.
For a finite Galois extension over a countable field, the Natarajan dimension equals the group-invariant coset-cube number, whose value is at least the number of complement factors of any chief series of any subgroup.
For every solvable Galois group, the DS dimension and the Natarajan dimension coincide exactly.
For the Galois groups A5 and S5 of the quintic equation, the DS dimension and the Natarajan dimension exhibit gaps of 2<3 and 3<4 respectively, and A5 is the smallest group for which such a gap exists.
For the Galois group Sn of a general equation of degree n, the DS dimension coincides with the Natarajan dimension if and only if n ≤ 4; in the characteristic-zero case, this is equivalent to the equation being solvable by radicals (the Abel-Ruffini theorem).
For linear, orthogonal and unitary maps on R^d or C^d, both the DS dimension and the Natarajan dimension equal d.
All Facts (6)
In the agnostic learning setting, the gap between the sample complexity of every ERM and the known lower bound is at most the ratio of the DS dimension to the Natarajan dimension multiplied by a constant.
50%UnverifiedFor a finite Galois extension over a countable field, the Natarajan dimension equals the group-invariant coset-cube number, whose value is at least the number of complement factors of any chief series of any subgroup.
50%UnverifiedFor every solvable Galois group, the DS dimension and the Natarajan dimension coincide exactly.
50%UnverifiedFor the Galois groups A5 and S5 of the quintic equation, the DS dimension and the Natarajan dimension exhibit gaps of 2<3 and 3<4 respectively, and A5 is the smallest group for which such a gap exists.
50%UnverifiedFor the Galois group Sn of a general equation of degree n, the DS dimension coincides with the Natarajan dimension if and only if n ≤ 4; in the characteristic-zero case, this is equivalent to the equation being solvable by radicals (the Abel-Ruffini theorem).
50%UnverifiedFor linear, orthogonal and unitary maps on R^d or C^d, both the DS dimension and the Natarajan dimension equal d.
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