The underlyingmetricis,however,non-parametric.Wedevelopamaximumlikelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, andprovidethecorresponding EMalgorithm.

A.1 Notationandpreliminaries We consider the metric space(X,d( , )) where d : X X R+. We consider`( , ) to be the cross-entropyloss `log(M(x),y), ylogσ(M(x)) (1 y)logσ((1 M(x))) (whereσ(x) = 11+exp( x) isthe sigmoid function) orthe`2 loss.

This implies that M(δ) = T(δ), i.e., the constraint(13) made in T(δ) does not lose any generality in matrix representation. One technical distinction relative to the previous works [2,3] arises from the fact that in our setting, the hamming distances(dx1(`),dx2(`),dx3(`)) defined w.r.t. We focus on the family of rating matrices{Mhci: c T`}. First, we present the following lemma that guarantees the existence of two subsets of users with certainproperties. The proof of this case follows the same structure as that of the grouping-limited regime. It is shown that the groups within each cluster are recovered with a vanishing fraction of errors if Ig = ω(1/n).

The matching lower bounds utilize several different Wasserstein distances.

The way to instantiate BACON will be similar to MFN. The following Lemma will showthat Definition 1.2 can be extended toanalyzing functions from differentdomain. Let F = gL g1 γ, with gi being a multivariate polynomial. The inductive hypothesis is: fork 1, if zk[j] is linear sum ofB for all j, then zk+1[l]islinearsumsofB foralll. By definition ofz, we know thatzk+1 = gk(zk), where gk is a multivariate polynomial of finite degreed.

Multiple-try Metropolis (MTM) is a popular Markov chain Monte Carlo method with the appealing feature of being amenable to parallel computing. At each iteration, it samples several candidates for the next state of the Markov chain and randomly selects one of them based on a weight function. The canonical weight function is proportional to the target density. We show both theoretically and empirically that this weight function induces pathological behaviours in high dimensions, especially during the convergence phase. We propose to instead use weight functions akin to the locally-balanced proposal distributions of Zanella (2020), thus yielding MTM algorithms that do not exhibit those pathological behaviours. To theoretically analyse these algorithms, we study the high-dimensional performance of ideal schemes that can be thought of as MTM algorithms which sample an infinite number of candidates at each iteration, as well as the discrepancy between such schemes and the MTM algorithms which sample a finite number of candidates. Our analysis unveils a strong distinction between the convergence and stationary phases: in the former, local balancing is crucial and effective to achieve fast convergence, while in the latter, the canonical and novel weight functions yield similar performance. Numerical experiments include an application in precision medicine involving a computationally-expensive forward model, which makes the use of parallel computing within MTM iterations beneficial.

We consider inference problems for a class of continuous state collective hidden Markov models, where the data is recorded in aggregate (collective) form generated by a large population of individuals following the same dynamics. We propose an aggregate inference algorithm called collective Gaussian forward-backward algorithm, extending recently proposed Sinkhorn belief propagation algorithm to models characterized by Gaussian densities. Our algorithm enjoys convergence guarantee. In addition, it reduces to the standard Kalman filter when the observations are generated by a single individual. The efficacy of the proposed algorithm is demonstrated through multiple experiments.