The total variation distance between high-dimensional Gaussians

We prove a lower bound and an upper bound for the total variation distance between two high-dimensional Gaussians, which are within a constant factor of one another.Comment: 12 page

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the computational problem of determining the TV distance between two product distributions over the domain {0, 1}n. We establish the following results. 1. Exact computation of TV distance between two product distributions is #P-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals. 2. Given two product distributions P and Q with marginals of P being at least 1/2 and marginals of Q being at most the respective marginals of P, there exists a fully polynomial-time randomized approximation scheme (FPRAS) for computing the TV distance between P and Q. In particular, this leads to an efficient approximation scheme for the interesting case when P is an arbitrary product distribution and Q is the uniform distribution. We pose the question of characterizing the complexity of approximating the TV distance between two arbitrary product distributions as a basic open problem in computational statistics

Optimal stopping time and halting set for total variation distance

An aperiodic and irreducible Markov chain on a finite state space converges to its stationary distribution. When convergence to equilibrium is measured by total variation distance, there exists an optimal coupling and a maximal coupling time. In this article, the maximal coupling time is compared to the hitting time of a specific state or set. Such sets, named halting sets, are studied in the case of symmetric birth-and-death chains and in some other examples. Some applications to the cutoff phenomenon are given. These results yield new methods to calculate cutoff times for some monotone birth-and death chains without the lazy hypothesis

Bounds for Approximation in Total Variation Distance by Quantum Circuits

It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and functions are exponentially hard to approximate [Knill 1995]. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e-\epsilon even for exponentially small \epsilon. The quantity 2/e is asymptotically the average distance to the uniform distribution. Almost all states with probability amplitudes concentrated in a small fraction of the space are hard to approximate to within a TVD of 2-\epsilon. These results imply that non-uniform quantum circuit complexity is non-trivial in any reasonable model. They also reinforce the notion that the relative information distance between states (which is based on the difficulty of transforming one state to another) fully reflects the dimensionality of the space of qubits, not the number of qubits.Comment: uuencoded compressed postscript, LACES 68Q-95-3

On Deterministically Approximating Total Variation Distance

Total variation distance (TV distance) is an important measure for the difference between two distributions. Recently, there has been progress in approximating the TV distance between product distributions: a deterministic algorithm for a restricted class of product distributions (Bhattacharyya, Gayen, Meel, Myrisiotis, Pavan and Vinodchandran 2023) and a randomized algorithm for general product distributions (Feng, Guo, Jerrum and Wang 2023). We give a deterministic fully polynomial-time approximation algorithm (FPTAS) for the TV distance between product distributions. Given two product distributions $\mathbb{P}$ and $\mathbb{Q}$ over $[q]^n$, our algorithm approximates their TV distance with relative error $\varepsilon$ in time $O\bigl( \frac{qn^2}{\varepsilon} \log q \log \frac{n}{\varepsilon \Delta_{\text{TV}}(\mathbb{P},\mathbb{Q}) } \bigr)$. Our algorithm is built around two key concepts: 1) The likelihood ratio as a distribution, which captures sufficient information to compute the TV distance. 2) We introduce a metric between likelihood ratio distributions, called the minimum total variation distance. Our algorithm computes a sparsified likelihood ratio distribution that is close to the original one w.r.t. the new metric. The approximated TV distance can be computed from the sparsified likelihood ratio. Our technique also implies deterministic FPTAS for the TV distance between Markov chains