text stringlengths 128 2.05k |
|---|
Cauchy Sequences for the Covariant Propinquity We now study the problem of convergence of Cauchy sequences for the covariant propinquity. We begin with the following corollary of 17 , Theorem 2.13] , which extends 17 , Theorem 3.10] to the proper setting we are now working within. This result encapsulates some of the c... |
and and in particular, we recall what a target set and a forward target set is. Let [MATH] and [MATH] be two quantum compact metric spaces. Let [MATH] be a tunnel from [MATH] to [MATH] . For any [MATH] and [MATH] , the [MATH] -target set of [MATH] is defined by: |
[EQUATION] Now, if [MATH] is a covariant tunnel, then for all [MATH] and [MATH] , by a mild abuse of notations, we write [MATH] for [MATH] where [MATH] |
Moreover, we denote [MATH] as [MATH] Now, by 11 , Corollary 4.5] 16 , Proposition 2.12] , if [MATH] and [MATH] , and if [MATH] and [MATH] then: |
[EQUATION] Let now [MATH] be a Lipschitz linear map. Let [MATH] be a tunnel from [MATH] to [MATH] . For [MATH] and [MATH] , the [MATH] -image-target set of [MATH] is defined by: |
[EQUATION] and the [MATH] -forward-target set of [MATH] , for [MATH] is defined by: [EQUATION] Now, by 17 , Lemma 2.5] , if [MATH] and [MATH] , and if [MATH] and [MATH] then: |
[EQUATION] As before, if [MATH] is a covariant tunnel, we write [MATH] for the forward target set associated to the underlying tunnel of [MATH] |
We now recall and mildly extend a metric introduced in Theorem 4.1 Let [MATH] be a quantum compact metric space and let [MATH] be a unital C*-algebra. If for any two unital linear maps [MATH] [MATH] from [MATH] to [MATH] , we set: |
[EQUATION] then [MATH] is a distance on the space [MATH] of unit preserving bounded linear maps, which, on any norm-bounded subset, metrizes the initial topology induced by the family of seminorms: |
[EQUATION] Proof. Let [MATH] be a sequence of unit preserving linear maps converging to some unital linear map [MATH] for [MATH] , and for which there exists some [MATH] such that for all [MATH] , we have [MATH] , where [MATH] is the operator norm for linear maps from [MATH] to [MATH] . Let [MATH] and [MATH] . Since [M... |
[EQUATION] Thus for all [MATH] , the sequence [MATH] converges to [MATH] . By linearity, we then conclude [MATH] converge to [MATH] for [MATH] |
Conversely, assume that for all [MATH] , the sequence [MATH] converges to [MATH] in [MATH] , and again assume that there exists [MATH] such that for all [MATH] , we have [MATH] . Let [MATH] and fix [MATH] . As [MATH] is a L-seminorm, [MATH] is totally bounded. Thus, there exists a finite [MATH] -dense set [MATH] of [MA... |
[EQUATION] Thus for [MATH] , we have [MATH] We now can prove: Theorem 4.2 Let [MATH] be a permissible function. Let [MATH] be a sequence of Lipschitz dynamical [MATH] -systems such that: |
1. [MATH] converges to some [MATH] -quantum compact metric spaces [MATH] for [MATH] 2. [MATH] converges to a proper monoid [MATH] for [MATH] |
3. there exists a locally bounded function [MATH] such that for all [MATH] and for all [MATH] , we have [MATH] 4. for all [MATH] there exists [MATH] and [MATH] such that for all [MATH] , if [MATH] and [MATH] then: |
[EQUATION] then there exist: 1. a strongly continuous action [MATH] of [MATH] on [MATH] such that [MATH] is a Lipschitz dynamical [MATH] -system, |
2. for all [MATH] , an almost isometry [MATH] such that [MATH] 3. a strictly increasing sequence [MATH] 4. for each [MATH] , a tunnel [MATH] from [MATH] to [MATH] , with |
[EQUATION] such that for all [MATH] and [MATH] , with [MATH] and [MATH] [EQUATION] In particular, for all [MATH] , we have [MATH] . Furthermore, for all [MATH] there exist [MATH] and [MATH] such that if [MATH] with [MATH] , if [MATH] and if [MATH] then [MATH] |
If moreover: 1. for all [MATH] , the action [MATH] is by Lipschitz morphisms, then [MATH] is also an action by Lipschitz morphisms, |
2. for all [MATH] , the action [MATH] is by Lipschitz automorphisms, then [MATH] is also an action by Lipschitz automorphisms, 3. |
for all [MATH] [MATH] is a group and the action [MATH] is by full quantum isometries, then [MATH] is also an action by full quantum isometries, |
4. for all [MATH] [MATH] is a compact group and [MATH] is an ergodic action by full quantum isometries, then [MATH] is also an ergodic action. |
Proof. For all [MATH] , let [MATH] and: [EQUATION] Since [MATH] is a proper metric space, it is separable. Let [MATH] be a countable dense subset of [MATH] containing the identity element [MATH] of [MATH] . Let [MATH] be the sub-monoid [MATH] . Since [MATH] consists of all the finite products of elements of the countab... |
Let [MATH] . By assumption, there exists [MATH] and [MATH] such that if [MATH] and [MATH] with [MATH] then [MATH] . Now, fix [MATH] . There exists [MATH] such that if [MATH] then [MATH] by Assertion (4) of Lemma ( 2.16 ) since [MATH] and [MATH] |
Let [MATH] with [MATH] . Let [MATH] with [MATH] . We now compute: [EQUATION] We also record that [MATH] , for [MATH] the identity element of [MATH] , is the identity map. |
As a monoid, [MATH] is trivially a semigroupoid over the set of a single object, which we take as the identity element of [MATH] . The domain and codomain maps [MATH] and [MATH] from [MATH] to [MATH] are obviously constant, and the multiplication on [MATH] is the composition operation on the semigroupoid [MATH] . Thus ... |
[EQUATION] where we use the notation [MATH] for [MATH] Note that by 17 , Theorem 2.13] , for all [MATH] , the linear map [MATH] is defined on [MATH] , is unital and positive, and moreover it is a unital *-endomorphism if [MATH] are actions by unital *-endomorphisms for all [MATH] , and even a *-automorphism if [MATH] i... |
Let [MATH] [MATH] [MATH] . Let [MATH] . There exists [MATH] and [MATH] such that for all [MATH] , if [MATH] and [MATH] then [MATH] . Let [MATH] such that [MATH] |
Let [MATH] such that for all [MATH] , we have [MATH] by Lemma ( 2.16 ). Since [MATH] is locally bounded, and since: [EQUATION] there exists [MATH] and [MATH] such that for all [MATH] we have [MATH] and [MATH] |
Let [MATH] such that for all [MATH] , we have: [EQUATION] Let [MATH] such that for all [MATH] , we have [MATH] Let [MATH] . Now let [MATH] . Let [MATH] and [MATH] . Let [MATH] and [MATH] . We then have: |
[EQUATION] Hence, [MATH] is uniformly continuous over a dense subset [MATH] of the complete space [MATH] . Hence, it admits a unique uniformly continuous extension to [MATH] , but we are going to prove a little more. By assumption, for all [MATH] , we have [MATH] where [MATH] is the operator norm for a linear map on [M... |
[EQUATION] Hence for all [MATH] , the map [MATH] is uniformly continuous on [MATH] , which is dense in the complete metric space [MATH] . Hence, it admits a unique equicontinuous extension to [MATH] , which we still denote by [MATH] |
Moreover, [MATH] is a unital positive [MATH] -linear map (hence, of norm [MATH] ) and if for all [MATH] , the action [MATH] is by Lipschitz endomorphism, then [MATH] is a Jordan-Lie morphism for all [MATH] |
Another consequence of this observation is that [MATH] is locally bounded. Indeed, let [MATH] . There exists [MATH] and [MATH] such that if [MATH] then [MATH] . Assume now [MATH] . There exists a sequence [MATH] in [MATH] converging to [MATH] — we may as well assume that [MATH] , and thus [MATH] for all [MATH] . We con... |
[EQUATION] for all [MATH] . Hence, [MATH] is locally bounded as well (though not a priori using the function [MATH] ). Setting, [MATH] for [MATH] and [MATH] , we check that [MATH] is a positive unital linear map (hence still of norm [MATH] ) on [MATH] , which is a unital *-endomorphism of [MATH] if [MATH] is a Jordan-L... |
It is also easy to check that [MATH] is a monoid action on [MATH] since [MATH] is, and since the multiplication on [MATH] is assumed continuous. By construction, the action [MATH] is strongly continuous on [MATH] and hence on [MATH] by an immediate computation. If [MATH] is a group, since [MATH] is the identity, we the... |
Now, we show that Expression ( 4.1 ) holds for all [MATH] rather than all [MATH] . Let [MATH] and [MATH] . Let [MATH] . Since both [MATH] and [MATH] are locally bounded, there exists [MATH] and [MATH] such that if [MATH] then [MATH] |
Let [MATH] . By uniform continuity of [MATH] , there exists [MATH] such that if [MATH] and [MATH] then [MATH] . By density of [MATH] , there exists [MATH] such that [MATH] |
Now, there exists [MATH] such that for all [MATH] , we have: [EQUATION] Moreover, there exists [MATH] such that if [MATH] then [MATH] , and [MATH] such that if [MATH] then [MATH] |
Let [MATH] Let [MATH] . There exists [MATH] such that: [EQUATION] Let [MATH] . Note that [MATH] . We now estimate: [EQUATION] Hence: |
[EQUATION] as desired. Now, Expression ( 4.1 ) holds for any [MATH] — since all forward target sets [MATH] have diameter converging to [MATH] and are not empty for [MATH] , and [MATH] with [MATH] — so we may apply, for instance, 19 , Claim 6.13] |
In turn, this proves that [MATH] by lower semi-continuity of [MATH] We make one last observation. Let [MATH] . There exist [MATH] and [MATH] such that if [MATH] [MATH] and [MATH] then [MATH] . For all [MATH] with [MATH] , then there exists [MATH] with [MATH] such that if [MATH] then [MATH] since [MATH] . By 17 , Theore... |
Let now [MATH] with [MATH] . By density of [MATH] , there exists two sequences [MATH] [MATH] such that [MATH] and [MATH] . Let [MATH] with [MATH] . We have: |
[EQUATION] This concludes our proof, as the remaining arguments regarding full quantum isometries and ergodicity follows as in 17 , Theorem 3.14] |
We now can state a sufficient condition for a certain kind of compactness. Theorem 4.3 Let [MATH] be an [MATH] -quantum compact metric space and [MATH] be a proper monoid. Let [MATH] be a sequence of Lipschitz dynamical systems and let [MATH] be a locally bounded function such that: |
1. for all [MATH] and [MATH] , we have [MATH] 2. [MATH] 3. [MATH] 4. for all [MATH] , there exists [MATH] and [MATH] such that if [MATH] and if [MATH] with [MATH] , then [MATH] |
then there exists a strictly increasing function [MATH] and a Lipschitz dynamical system [MATH] such that: [EQUATION] As the action [MATH] is given by Theorem ( 4.2 ), it enjoys the properties described in the conclusion of that theorem. |
Proof. By Theorem ( 4.2 ), there exists a strongly continuous action [MATH] of [MATH] on [MATH] , a strictly increasing function [MATH] and a sequence: |
[EQUATION] of tunnels where [MATH] is a tunnel from [MATH] to [MATH] for all [MATH] , such that [MATH] is a Lipschitz dynamical [MATH] -system and for all [MATH] [MATH] , and [MATH] [MATH] |
[EQUATION] By replacing the original sequences of tunnels, Lipschitz dynamical systems, and almost isometries by their subsequence indexed by [MATH] , we will dispense with writing [MATH] in the rest of this proof. We also write [MATH] for all [MATH] and [MATH] |
Let [MATH] . By assumption, there exists [MATH] and [MATH] such that for all [MATH] and [MATH] , we have [MATH] then [MATH] Fix [MATH] . By compactness, there exists a [MATH] -dense finite subset [MATH] (for [MATH] ) of [MATH] . As [MATH] is proper, the closed ball [MATH] is compact, so there exists a finite [MATH] -de... |
[EQUATION] Note that [MATH] is finite by construction. Since [MATH] is compact, and since [MATH] is locally bounded, we conclude that there exists [MATH] such that [MATH] for all [MATH] |
There exists [MATH] such that for all [MATH] [EQUATION] for all [MATH] and [MATH] There exists [MATH] such that for all [MATH] , we have [MATH] |
Let [MATH] . There exists [MATH] such that for all [MATH] , we have that: [EQUATION] Fix [MATH] Let [MATH] . There exists [MATH] such that [MATH] . (We note that all the computations below are also valid if we start with [MATH] and obtain [MATH] such that [MATH] ). |
Let [MATH] with [MATH] and [MATH] . Let [MATH] . Write [MATH] and [MATH] — note that [MATH] . First, by definition of [MATH] and [MATH] , there exists [MATH] such that: |
[EQUATION] and [MATH] such that [MATH] , and therefore [MATH] , which leads to: [EQUATION] Let now [MATH] . We compute: [EQUATION] |
Therefore: [EQUATION] Similarly, we also have: [EQUATION] and therefore: [EQUATION] Note that [MATH] by definition. Therefore, [MATH] . Let [MATH] . By construction, [MATH] and thus [MATH] . Moreover, by definition of target set, there exists [MATH] such that [MATH] and [MATH] and [MATH] |
[EQUATION] Thus, for all [MATH] , for all [MATH] , and for all [MATH] with [MATH] and [MATH] , we have proven: [EQUATION] Now, let [MATH] and [MATH] . Again, there exists [MATH] such that [MATH] . By our previous work, since [MATH] by Lemma ( 2.16 ), we note that: |
[EQUATION] Therefore, we conclude: [EQUATION] Now, let [MATH] with [MATH] . Then we note that [MATH] and [MATH] as L-seminorms vanish on scalars, and thus we have just proven: |
[EQUATION] Our computation is again valid if we start with [MATH] and choose [MATH] with [MATH] Therefore [MATH] . Since [MATH] , we conclude that [MATH] . Hence we have completed our proof. |
We thus can conclude on a sufficient condition for convergence of Cauchy sequences for the covariant propinquity: Corollary 4.4 Let [MATH] be permissible and continuous and let [MATH] be a locally bounded function. Let [MATH] be a sequence of Lipschitz dynamical [MATH] -systems and [MATH] a sequence of positive real nu... |
1. [MATH] 2. for all [MATH] and [MATH] [EQUATION] 3. [MATH] 4. [MATH] 5. for all [MATH] , there exists [MATH] and [MATH] such that if [MATH] and if [MATH] with [MATH] , then [MATH] |
then there exists a Lipschitz dynamical [MATH] -system [MATH] such that: [EQUATION] Moreover, if for all [MATH] , the action [MATH] is by *-endomorphisms, then so is the action [MATH] |
Proof. The sequence [MATH] is a Cauchy sequence for the dual [MATH] -propinquity, which is complete by since [MATH] is continuous, so there exists a quantum compact metric space [MATH] such that [MATH] Moreover by Theorem ( 3.4 ), there exists a proper monoid [MATH] such that: |
[EQUATION] By Theorem ( 4.2 ), which we now may apply, there exists a strictly increasing function [MATH] such that: [EQUATION] converges to a Lipschitz dynamical [MATH] -system [MATH] |
Since a Cauchy sequence with a convergent subsequence converges, our corollary is now proven. A particular consequence of our work is a simpler result concerning Lipschitz dynamical systems with bi-invariant metrics, since regularity is no longer an hypothesis. |
Corollary 4.5 Let [MATH] be a sequence of Lipschitz dynamical systems with [MATH] bi-invariant for all [MATH] . If: 1. [MATH] is Cauchy for [MATH] |
2. there exists a locally bounded function [MATH] such that: [EQUATION] where [MATH] is the identity element of [MATH] for all [MATH] |
3. for all [MATH] , there exists [MATH] and [MATH] such that if [MATH] and if [MATH] with [MATH] , then [MATH] then there exists a Lipschitz dynamical system [MATH] such that: |
[EQUATION] Moreover, if for all [MATH] , the action [MATH] is by *-endomorphisms, then so is the action [MATH] We also record the implications of our work on Lipschitz C*-dynamical systems, under the strong assumption of Corollary ( 3.6 ). |
Corollary 4.6 Let [MATH] be permissible and continuous. If [MATH] is a Cauchy sequence of Lipschitz C*-dynamical [MATH] -systems such that: |
1. for all [MATH] there exists [MATH] and [MATH] such that for all [MATH] , if [MATH] with [MATH] then [MATH] 2. for all [MATH] , there exists [MATH] and [MATH] such that if [MATH] and if [MATH] with [MATH] , then [MATH] |
3. there exists a locally bounded function [MATH] such that for all [MATH] [MATH] , and with [MATH] the identity of [MATH] , we have [MATH] |
then there exists a Lipschitz C*-dynamical [MATH] -system [MATH] such that: [EQUATION] Proof. This follows from Theorem ( 4.3 ) and Corollary ( 3.6 ). |
We conclude this section with two observations. First, there are many natural complete classes of Lipschitz dynamical systems for the covariant propinquity. Let [MATH] be a continuous admissible function, [MATH] be a locally bounded function, and [MATH] be a function with [MATH] and [MATH] continuous at [MATH] . Let us... |
1. [MATH] is a proper monoid with [MATH] bi-invariant, 2. for all [MATH] , we have [MATH] 3. for all [MATH] [MATH] (where [MATH] is the identity element of [MATH] ). |
Let [MATH] be the subclass of [MATH] consisting of Lipschitz C*-dynamical systems. Then Theorem ( 4.3 ) and its corollaries prove that both [MATH] and [MATH] are complete for [MATH] . These are but certain possible complete classes: for instance, we could relax the hypothesis of working with bi-invariant metrics by ask... |
Second, there is a natural way to metrize compact monoids and groups acting on a quantum compact metric space. We discuss this point in the case of compact groups. Let us be given a compact group [MATH] and a strongly continuous action [MATH] of [MATH] by Lipschitz automorphisms of a quantum compact metric space [MATH]... |
[EQUATION] In general, [MATH] may only be a pseudo-metric, though it is induced by a pseudo-length function [MATH] (where [MATH] is the identity automorphism of [MATH] ). We note that [MATH] is a continuous function since [MATH] metrizes the topology of pointwise convergence in norm on the group of automorphisms of [MA... |
and since [MATH] is strongly continuous. Now, suppose [MATH] for some [MATH] , then for any [MATH] we note that: [EQUATION] so [MATH] is closed by all inner automorphisms of [MATH] . It is then easy to check that [MATH] is a subgroup, hence a normal subgroup, and it is closed by continuity of [MATH] . As a consequence,... |
# Source: arxiv 1806.04743 # Title: INFERNO: Inference-Aware Neural Optimisation # Sections: all # Downloaded: 2026-03-03T05:16:05.575741+00:00 |
INFERNO: Inference-Aware Neural Optimisation Abstract Complex computer simulations are commonly required for accurate data modelling in many scientific disciplines, making statistical inference challenging due to the intractability of the likelihood evaluation for the observed data. Furthermore, sometimes one is intere... |
Introduction Simulator-based inference is currently at the core of many scientific fields, such as population genetics, epidemiology, and experimental particle physics. In many cases the implicit generative procedure defined in the simulation is stochastic and/or lacks a tractable probability density [MATH] , where |
[MATH] is the vector of model parameters. Given some experimental observations [MATH] , a problem of special relevance for these disciplines is statistical inference on a subset of model parameters |
[MATH] This can be approached via likelihood-free inference algorithms such as Approximate Bayesian Computation (ABC) , simplified synthetic likelihoods |
or density estimation-by-comparison approaches Because the relation between the parameters of the model and the data is only available via forward simulation, most likelihood-free inference algorithms tend to be computationally expensive due to the need of repeated simulations to cover the parameter space. When data ar... |
are used instead of the raw data for tractability. The choice of summary statistics for such cases becomes critical, given that naive choices might cause loss of relevant information and a corresponding degradation of the power of resulting statistical inference. |
As a motivating example we consider data analyses at the Large Hadron Collider (LHC), such as those carried out to establish the discovery of the Higgs boson \autocites higgs2012cmshiggs2012atlas. In that framework, the ultimate aim is to extract information about Nature from the large amounts of high-dimensional data ... |
[MATH] cannot be analytically computed. The inference problem in particle physics is commonly posed as hypothesis testing based on the acquired data. An alternate hypothesis |
[MATH] (e.g. a new theory that predicts the existence of a new fundamental particle) is tested against a null hypothesis [MATH] (e.g. an existing theory, which explains previous observed phenomena). The aim is to check whether the null hypothesis can be rejected in favour of the alternate hypothesis at a certain confid... |
Due to the high dimensionality of the observed data, a low-dimensional summary statistic has to be constructed in order to perform inference. A well-known result of classical statistics, the Neyman-Pearson lemma |
, establishes that the likelihood-ratio [MATH] is the most powerful test when two simple hypotheses are considered. As [MATH] and [MATH] are not available, simulated samples are used in practice to obtain an approximation of the likelihood ratio by casting the problem as supervised learning classification. |
In many cases, the nature of the generative model (a mixture of different processes) allows the treatment of the problem as signal (S) vs background (B) classification |
, when the task becomes one of effectively estimating an approximation of [MATH] which will vary monotonically with the likelihood ratio. While the use of classifiers to learn a summary statistic can be effective and increase the discovery sensitivity, the simulations used to generate the samples which are needed to tr... |
In this work, we present a new machine learning method to construct non-linear sample summary statistics that directly optimises the expected amount of information about the subset of parameters of interest using simulated samples, by explicitly and directly taking into account the effect of nuisance parameters. In add... |
Problem Statement Let us consider a set of [MATH] i.i.d. observations [MATH] where [MATH] , and a generative model which implicitly defines a probability density |
[MATH] used to model the data. The generative model is a function of the vector of parameters [MATH] which includes both relevant and nuisance parameters. We want to learn a function |
[MATH] that computes a summary statistic of the dataset and reduces its dimensionality so likelihood-free inference methods can be applied effectively. From here onwards, [MATH] will be used to denote the dimensionality of the summary statistic [MATH] |
While there might be infinite ways to construct a summary statistic [MATH] , we are only interested in those that are informative about the subset of interest |
[MATH] of the model parameters. The concept of statistical sufficiency is especially useful to evaluate whether summary statistics are informative. In the absence of nuisance parameters, classical sufficiency can be characterised by means of the factorisation criterion: |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.