We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove several new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds.

Our main result is concerned with testing identity to a specific distribution $p$, given as a parameter. In a recent and influential work, Valiant and Valiant [VV14] showed that the sample complexity of the aforementioned problem is closely related to the $\ell_{2/3}$-quasinorm of $p$. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs.
More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's $K$-functional. We show that this quantity is closely related to the size of the effective support of $p$ (loosely speaking, the number of supported elements that constitute the vast majority of the mass of $p)$.
This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.

Fixed minor typos, updated references.

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove several new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds.

Our main result is concerned with testing identity to a specific distribution $p$, given as a parameter. In a recent and influential work, Valiant and Valiant [VV14] showed that the sample complexity of the aforementioned problem is closely related to the $\ell_{2/3}$-quasinorm of $p$. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs.
More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's $K$-functional. We show that this quantity is closely related to the size of the effective support of $p$ (loosely speaking, the number of supported elements that constitute the vast majority of the mass of $p)$.
This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.