  Score matching is an alternative to maximum likelihood (ML) for estimating a
probability distribution parametrized up to a constant of proportionality. By
fitting the ''score'' of the distribution, it sidesteps the need to compute
this constant of proportionality (which is often intractable). While score
matching and variants thereof are popular in practice, precise theoretical
understanding of the benefits and tradeoffs with maximum likelihood -- both
computational and statistical -- are not well understood. In this work, we give
the first example of a natural exponential family of distributions such that
the score matching loss is computationally efficient to optimize, and has a
comparable statistical efficiency to ML, while the ML loss is intractable to
optimize using a gradient-based method. The family consists of exponentials of
polynomials of fixed degree, and our result can be viewed as a continuous
analogue of recent developments in the discrete setting. Precisely, we show:
(1) Designing a zeroth-order or first-order oracle for optimizing the maximum
likelihood loss is NP-hard. (2) Maximum likelihood has a statistical efficiency
polynomial in the ambient dimension and the radius of the parameters of the
family. (3) Minimizing the score matching loss is both computationally and
statistically efficient, with complexity polynomial in the ambient dimension.
