SlowGuess/NeurIPS_2023_OCR huggingface dataset:
As the reader may have noticed already, there are two equivalence relations at play on  $X$ , namely  $\sim$  and  $\approx$ . Although the construction of the space  $X / \approx$  is dependant on the equivalence  $\sim$  through the definition of  $d_1$ , it is not straightforward to identify the relationship between the two abstract quotient spaces  $X / \sim$  and  $X / \approx$ . Now is a good time to pause and unveil the primary reason to justify the development of these technical tools. Our initial goal is to metrize the topological quotient  $X / \sim$  for a given equivalence relation  $\sim$  so that we could extend Wald's conditions [Wald, 1949] in  $X / \sim$  to prove consistency results in this space. The procedure outlined in the previous paragraph provides a recipe for constructing a metric  $\bar{d}$  on the space  $X / \approx$ . The next crucial result, Lemma 4.3 shows that for certain nice equivalence relations  $\sim$ , the two spaces  $X / \approx$  and  $X / \sim$  are identical. This result is a key fact that allows us to maneuver the metric construction into  $X / \sim$ . We now state the result from Weaver [2018]. The proof of this result is skipped as it is deemed to be technical and it serves a little purpose to our main goal. For a thorough treatment of metrizability of quotient topological spaces, we refer the inquisitive reader to the comprehensive book on Lipschitz algebras [Weaver, 2018].
Lemma 4.3. If  $C$  is a closed subset of a complete metric space  $X$ . Let  $X / C$  denote the quotient space of  $X$  by the equivalence defined by  $x \sim y$  if either  $x = y$  or  $x, y \in C$  then the underlying set  $X_{\approx}$  and the quotient  $X / C$  are the same. The metric on  $X / C$  in this case, will take the form:
$$
\bar {d} ([ x ], [ y ]) = \min  (d (x, y), d (x, C) + d (y, C)) \quad f o r a n y [ x ], [ y ] \in X / C.
$$
In the above statement,  $d(x,C)$  denotes the distance between the point  $x$  and the closed subset  $C$  defined by  $d(x,C) := \inf_{c\in C}d(x,c)$  and likewise for  $d(y,C)$ . An immediate follow-up question of the above result would be: what is the space  $X$  and the closed subset  $C$  we use in our context? We now recall two things from the previous section, first the parameter space for the PPCA model which is  $\Theta \subset \mathbb{R}^{pq + 1}$  and second the issue of identifiability, which arose because there could exist two parameters  $\theta ,\phi \in \Theta$  such that their corresponding densities are equal, which is equivalent to  $\mathbf{W}_{\theta}\mathbf{W}_{\theta}^{T} + \sigma_{\theta}^{2}I_{p} = \mathbf{W}_{\phi}\mathbf{W}_{\phi}^{T} + \sigma_{\phi}^{2}I_{p}$ . In what follows, using Redner's idea [Redner, 1981] we consider the closed subset defined by
$$
C := \left\{\theta \in \Theta : \mathbf {W} _ {\theta} \mathbf {W} _ {\theta} ^ {T} + \sigma_ {\theta} ^ {2} I _ {p} = \mathbf {W} _ {0} \mathbf {W} _ {0} ^ {T} + \sigma_ {0} ^ {2} I _ {p} \right\} \subset \Theta ,
$$
which allows us to get rid of identifiability since in the quotient topological space  $\Theta / C$ ,  $f(x; [\theta]) = f(x; [\theta_0])$  if and only if  $[\theta] = [\theta_0]$  or equivalently  $\theta \sim \theta_0$  by construction. We can now hope for checking Wald's consistency conditions in  $\Theta / C$  to prove  $[\hat{\theta}]$  converges to  $[\theta_0]$  in probability, where  $\hat{\theta} = (\hat{W}, \hat{\sigma}^2)$  denotes the MLE.
In the introduction, we pointed out that despite the extension of Wald's work [Wald, 1949] to quotient topological spaces by Redner [1981], an inadequate amount of attention was paid to the comprehensive understanding of the interplay between the quotient topology and the topology generated by the metric  $\bar{d}$  in that treatment. For instance, a priori it is, by no means obvious how one would interpret their assumption 3 in the quotient space  $\Theta / C$  without invoking Lemma 4.3. Also, their assumption 5 does not clearly mention what mode of convergence has been used since there are two available notions of convergence, one coming from the quotient topology and another coming from the metric topology generated by  $\bar{d}$ , and those two topologies are not same in general. It is a nontrivial fact that the latter is contained in the former (for a discussion on this topic see James [1990]). However, in most statistical applications, for example in the applications to mixture distributions and clustering [Redner, 1981], we tend to quotient the parameter space by a suitable closed subspace depending on the context, and subsequently the quotient metric takes an easier form as given in Lemma 4.3. Therefore, several technical constructions get greatly simplified in these applications.
In this section, we state our main results. Let  $\{x_i\}_{i=1}^n \subset \mathbb{R}^p$  be a sequence of data points that are generated from latent observations  $\{z_i\}_{i=1}^n \in \mathbb{R}^q$  via the PPCA model defined in (3.1) and recall from section 3 that the data points are assumed to be generated from the true parameter  $\theta_0 := (\mathbf{W}_0, \sigma_0^2) \in \Theta$  of the underlying model. Our data points  $x_i$  can therefore be viewed as a sequence of realizations of i.i.d. random variables whose true marginal density is given by  $f(x; \theta_0) = \mathcal{N}(x; 0, \mathbf{W}_0 \mathbf{W}_0^T + \sigma_0^2 I_p)$ . We note a crucial point that for two parameters  $\theta, \phi \in \Theta$ ,  $f(x; \theta) = f(x; \phi)$  holds if and only if  $[\theta] = [\phi]$  in  $\Theta / C$ . Therefore, we write  $f(x; \theta)$  for simplicity, instead of cumbersome  $f(x; [\theta])$  when stating results in the quotient space  $\Theta / C$ . In the spirit of Wald [1949] and Redner [1981], two of our main results are the following:
Theorem 5.1. Let  $S$  is any closed subset of  $\Theta$  not intersecting  $C$  then
$$
\mathbb {P} \left(\lim  _ {n} \sup  _ {\theta \in S} \prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta)}{f (x _ {i} ; \theta_ {0})} = 0\right) = 1.
$$
Theorem 5.2. Let  $\theta_{n}(x_{1},\ldots ,x_{n})$  be any measurable function of the observations  $x_{1},\ldots ,x_{n}$  such that
$$
\prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta_ {n})}{f (x _ {i} ; \theta_ {0})} \geqslant c > 0 f o r a l l n,
$$
then  $\mathbb{P}\left(\lim_{n}[\theta_n] = [\theta_0]\right) = 1$ , that is  $\theta_{n}$  converges to  $\theta_0$  in  $\theta /C$  almost surely, and therefore in probability.
The above results have elegant geometric interpretations. In particular, Theorem 5.1 implies that if  $U_{C}$  is any open neighbourhood containing  $\bar{C}$  then almost surely all but finitely many terms of the sequence  $\{\theta_n\}$  land inside  $U_{C}$ . As an immediate consequence of Theorem 5.2 when  $c = 1$ , we obtain:
Corollary 5.3 (Strong consistency of the MLE in the quotient space). Let  $\theta_{n}$  denote the sequence of maximum likelihood estimates of the PPCA model. We then have that
$$
\mathbb {P} \left(\lim  _ {n} [ \theta_ {n} ] = [ \theta_ {0} ]\right) = 1
$$
that is maximum likelihood estimates converge to the true parameter  $\theta_0$  almost surely, and therefore in probability in the quotient space  $\Theta / C$ .
Additionally, using the above corollary we obtain that the covariance for the PPCA process can be consistently (strong) estimated under a compactness assumption.
Theorem 5.4. Let  $\Theta_0$  be a compact subset of  $\Theta$  containing the point  $\theta_0 = (W_0, \sigma_0^2)$ . Let  $\theta_n = (\widehat{W}, \widehat{\sigma}^2)$  denote the sequence of maximum likelihood estimates of the PPCA model. Then we have
$$
\mathbb {P} \left(\widehat {\boldsymbol {W}} \widehat {\boldsymbol {W}} ^ {T} + \widehat {\sigma} ^ {2} I _ {p} \rightarrow \boldsymbol {W} _ {0} \boldsymbol {W} _ {0} ^ {T} + \sigma_ {0} ^ {2} I _ {p}\right) = 1.
$$
Now we talk about possible generalizations above results in light of Wolfowitz [1949], where the observations  $\{X_i\}_{i=1}^n$  are assumed to be identically distributed but need not be independent. As noted in Wolfowitz's work that Wald [1949] proves strong consistency (i.e. almost sure convergence) and his proof techniques can be extended for dependent random variables  $\{X_i\}_{i=1}^n$  as long as the sequence  $\{X_i\}$  satisfies the strong law of large numbers. In the spirit of Wald's work, Wolfowitz [1949] proves the consistency of ML estimates (i.e. convergence in probability) only under the assumption that the sequence  $\{X_i\}$  satisfies weak law of large numbers. The assumption that the sequence  $\{X_i\}$  satisfies the weak law of large numbers may seem technical but it offers a great deal of flexibility in real life applications due to the following result of Bernstein [Cacoullos, 2012].
Theorem 5.5 (Bernstein). Let  $\{X_i\}$  be a sequence of centered random variables. If there exists a constant  $\kappa > 0$  such that for every  $i \in \mathbb{N}$  we have  $\operatorname{Var}(X_i) \leqslant \kappa$  and if the following condition is true:
$$
\lim_{|i - j|\to \infty}Cov(X_{i},X_{j}) = 0,
$$
then the weak law of large numbers holds.
We may note that the above result could be extremely relevant in real life applications as intuitively it accommodates all statistical models which allows local dependence among observations, in the sense that  $X_{i}$  can significantly correlate with  $X_{j}$  as long as  $j$  remains sufficiently close to  $i$  (i.e.,  $|i - j|$  is small), but the correlation gradually vanishes as  $|i - j|$  becomes large. Temporal stochastic processes or time series data with appropriate assumptions could be tailored into the category of those statistical models where Bernstein's result applies. We therefore state our next set of results on the consistency of maximum likelihood estimates of the PPCA model but only under a weak law assumption. The following result is an immediate consequence of Wolfowitz [1949].
Theorem 5.6. Let the data points  $\{x_i\}_{i=1}^n$  be realizations of identically distributed random variables  $\{X_i\}_{i=1}^n$  whose density is given by  $f(x; \theta_0)$ . Let the sequence of random variables  $\{X_i\}$  satisfy the weak law of large numbers. Given  $\eta > 0$  and a closed subset  $S$  of  $\Theta$  not intersecting  $C$ , there exists a quantity  $h(S) \in (0,1)$  which depends only on the closed set  $S$  and an integer  $N(\eta, S)$  such that
$$
\mathbb {P} \left(\sup  _ {\theta \in S} \prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta)}{f (x _ {i} ; \theta_ {0})} > h ^ {n} (S)\right) <   \eta f o r e v e r y n \geqslant N (\eta , S).
$$
Theorem 5.7. Let the data points  $\{x_{i}\}_{i = 1}^{n}$  be realizations of identically distributed random variables  $\{X_i\}_{i = 1}^n$  whose density is given by  $f(x;\theta_0)$ . Let the sequence of random variables  $\{X_i\}$  satisfy the weak law of large numbers. Let  $\theta_n(x_1,\ldots ,x_n)$  be any measurable function of the observations  $x_{1},\ldots ,x_{n}$  such that
then  $[\theta_n] \xrightarrow{\mathbb{P}} [\theta_0]$ , that is  $\theta_{n}$  converges to  $\theta_0$  in  $\theta /C$  in probability.
Furthermore we also have the two following results in spirit of corollary 5.3 and Theorem 5.4.
Corollary 5.8 (Consistency of the MLE in the quotient space). Let the data points  $\{x_{i}\}_{i = 1}^{n}$  be realizations of identically distributed random variables  $\{X_i\}_{i = 1}^n$  whose density is given by  $f(x;\theta_0)$ . Let the sequence of random variables  $\{X_i\}$  satisfy the weak law of large numbers. Let  $\theta_{n}$  denote the sequence of maximum likelihood estimates of the PPCA model. We then have that
$$
[ \theta_ {n} ] \xrightarrow {\mathbb {P}} [ \theta_ {0} ],
$$
that is maximum likelihood estimates converge to the true parameter  $\theta_0$  in probability in the quotient space  $\Theta /C$ .
Additionally, using the above corollary we can obtain prove consistent covariance estimation is possible for the PPCA model under a compactness assumption.
Theorem 5.9 (Consistent estimation of covariance). Let the data points  $\{x_i\}_{i=1}^n$  be realizations of identically distributed random variables  $\{X_i\}_{i=1}^n$  whose density is given by  $f(x; \theta_0)$ . Let the sequence of random variables  $\{X_i\}$  satisfy the weak law of large numbers. Let  $\Theta_0$  be a compact subset of  $\Theta$  containing the point  $\theta_0 = (\mathbf{W}_0, \sigma_0^2)$ . Let  $\theta_n = (\widehat{\mathbf{W}}, \widehat{\sigma}^2)$  denote the sequence of maximum likelihood estimates of the PPCA model. Then we have
$$
\widehat {\boldsymbol {W}} \widehat {\boldsymbol {W}} ^ {T} + \widehat {\sigma} ^ {2} I _ {p} \stackrel {{\mathbb {P}}} {\rightarrow} \boldsymbol {W} _ {0} \boldsymbol {W} _ {0} ^ {T} + \sigma_ {0} ^ {2} I _ {p}.
$$
6 Proof of theoretical results
We only present the proofs of Theorem 5.7 and Theorem 5.9. Corollary 5.8 follows from Theorem 5.7. The proofs of Theorem 5.2, corollary 5.3 and Theorem 5.4 will be analogous.
Proof of Theorem 5.7. We fix two quantities  $\eta_1, \eta_2 > 0$ . We need to show that there exists  $N_0(\eta_1, \eta_2)$  such that
$$
\mathbb {P} \left(\bar {d} ([ \theta_ {n} ], [ \theta_ {0} ]) > \eta_ {1}\right) \leqslant \eta_ {2}
$$
holds for all  $n \geqslant N_0(\eta_1, \eta_2)$ . We consider the following closed subset of the parameter space  $\Theta$  which does not intersect  $C$
$$
S _ {0} := \left\{\theta \in \Theta : \bar {d} ([ \theta ], [ \theta_ {0} ]) \geqslant \eta_ {1} \right\}.
$$
Using Theorem 5.6 we select  $0 < h(\eta_2) < 1$  such that
$$
\mathbb {P} \left(\sup  _ {\theta \in S _ {0}} \prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta)}{f (x _ {i} ; \theta_ {0})} > h ^ {n} (\eta_ {2})\right) <   \eta_ {2} \text {f o r e v e r y} n \geqslant N (\eta_ {1}, \eta_ {2}),
$$
for some  $N(\eta_1, \eta_2)$ . The dependence on  $\eta_1$  comes from the choice of the set  $S_0$ . Next, we choose  $N_0$  such that  $h^n(\eta_2) < c$  holds for every  $n \geqslant N_0$ . Let  $N_0(\eta_1, \eta_2) = \max(N_0, N(\eta_1, \eta_2))$ . Therefore for all  $n \geqslant N_0(\eta_1, \eta_2)$ , we have that
$$
\begin{array}{l} \mathbb {P} (\bar {d} ([ \theta_ {n} ], [ \theta_ {0} ]) > \eta_ {1}) \leqslant \mathbb {P} \left(\sup  _ {\theta \in S _ {0}} \prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta)}{f (x _ {i} ; \theta_ {0})} \geqslant c\right) \\ \leqslant \mathbb {P} \left(\sup  _ {\theta \in S _ {0}} \prod_ {i = 1} ^ {n} \frac {f (x _ {i} ; \theta)}{f (x _ {i} ; \theta_ {0})} > h ^ {n} (\eta_ {2})\right) \\ \leqslant \eta_ {2}. \\ \end{array}
$$
To prove Theorem 5.9 we need the following lemma.
Lemma 6.1. Let  $Y$  be a complete metric space, and let  $\psi : \Theta \to Y$  be a Lipschitz map such that  $\psi(\theta) = \psi(\theta')$  whenever  $\theta \sim \theta'$ . Then  $\psi$  lifts to an unique Lipschitz map from  $\Theta / C$  to  $Y$ .
Proof. The proof of the statement follows from proposition 1.4.4 and proposition 1.4.3 in Weaver [2018].  $\square$
Proof of Theorem 5.9. We consider the function  $\psi : \Theta \to \mathbb{R}^{p \times p}$  defined by  $\psi(\mathbf{W}, \sigma^2) = \mathbf{W}\mathbf{W}^T + \sigma^2 I_p$ . Since  $\theta_0 \in \mathrm{int}(\Theta_0)$ , we can therefore consider the restriction  $\psi|_{\Theta_0}: \Theta_0 \to \mathbb{R}^{p \times p}$ , which is a Lipschitz function as  $\psi$  is  $C^1$  and  $\Theta_0$  is compact. Therefore the lift  $\tilde{\psi}|_{\Theta_0}: \Theta_0 / C \to \mathbb{R}^{p \times p}$  is Lipschitz and hence continuous with respect to the topology generated by the metric  $\bar{d}$ .
Invoking corollary 5.8, we have  $[\widehat{\mathbf{W}},\widehat{\sigma}^2 ]\xrightarrow{\mathbb{P}}[\mathbf{W}_0,\sigma_0^2 ]$ . Since  $\tilde{\psi}\big|_{\Theta_0}$  is continuous with respect to the topology generated by the quotient metric  $\bar{d}$  from the previous lemma, using standard results from probability theory we infer that  $\tilde{\psi}\big|_{\Theta_0}([\widehat{\mathbf{W}},\widehat{\sigma}^2])\xrightarrow{\mathbb{P}}\tilde{\psi}\big|_{\Theta_0}([\mathbf{W}_0,\sigma_0^2])$
As we saw in the demonstration above, Lemma 6.1 plays a crucial role. We would like to emphasize that Theorem 4.2 would not be sufficient to ensure the continuity of the lift of  $\psi$  with respect to the metric  $\bar{d}$  as the quotient topology needs to be same that of the topology generated by the metric  $\bar{d}$  on the quotient space  $\Theta /C$  (see the counterexamples subsection in the appendices).
7 Auxiliary Lemmas
Our goal for this section will be to state Wald's conditions [Wald, 1949] to establish the consistency of MLE of the PPCA model and interpret those in the quotient parameter space  $\Theta /C$  as stated in Redner [1981]. Unless otherwise stated, throughout this section we will be working in the quotient parameter space  $\Theta /C$  introduced in section 4. We recall that  $\theta_0 = (\mathbf{W}_0,\sigma_0^2)$  denotes the unknown parameter such that the true marginal distribution of the data points  $\{x_{i}\}_{i = 1}^{n}$  is given by the density  $f(x;\theta_0) = \mathcal{N}(0,\mathbf{W}_0\mathbf{W}_0^T +\sigma_0^2 I_p)$ . For a given  $r > 0$ , we let  $N_r([\theta])$  denote the closed ball of radius  $r$  around  $[\theta]$  in  $\Theta /C$ . Following Redner [1981] we begin by introducing the quantities:
$$
f (x, \theta , r) = \sup  _ {[ \phi ] \in N _ {r} ([ \theta ])} f (x; \phi) \text {a n d} f ^ {*} (x, \theta , r) = \max  \{1, f (x, \theta , r) \},
$$
$$
h (x, s) = \sup  _ {[ \phi ] \notin N _ {s} ([ \theta_ {0} ])} f (x; \phi) \text {a n d} h ^ {*} (x, s) = \max  \{1, h (x, s) \}.
$$
We are now in a position to state Wald's conditions as a series of following lemmas whose proofs will imply Theorem 5.1 and Theorem 5.6 as a consequence of the results stated in Wald [1949] and Wolfowitz [1949], respectively.
Lemma 7.1. The parameter space  $(\Theta / C, \bar{d})$  is a metric space with the property that every closed and bounded subset of  $\Theta / C$  is compact.
Lemma 7.2. For each parameter  $[\theta] \in \Theta / C$  and for sufficiently small  $r$  and sufficiently large  $s$ ,  $f(., \theta, r)$  is measurable and the following expectations are bounded
$$
\mathbb {E} _ {\theta_ {0}} \left[ \log f ^ {*} (x, \theta , r) \right] <   \infty a n d \mathbb {E} _ {\theta_ {0}} \left[ \log h ^ {*} (x, s) \right] <   \infty .
$$
Lemma 7.3. Let  $\{[\theta_i]\} \subset \Theta / C$  be a sequence. If  $\bar{d}([\theta_i], [\theta_0]) \to \infty$  then  $f(x; \theta_i) \to 0$  except on a  $P_{\theta_0}$ -null set which does not depend on the sequence  $\{[\theta_i]\}$ .
Lemma 7.4. For each  $([\theta ],[\phi ])\in \Theta /C\times \Theta /C$  We have
$$
\mathbb {E} _ {\phi} [ | \log f (x; \theta) | ] <   \infty .
$$
Lemma 7.5. If  $[\theta_i] \to [\theta]$  in  $\Theta / C$  then  $f(x; \theta_i) \to f(x; \theta)$  except on a  $P_{\theta_0}$ -null set which does not depend on the sequence  $\{[\theta_i]\}$ .
The above conditions are fairly technical and primarily deal with the geometry of the quotient parameter space  $\Theta / C$ . One also needs to be careful as there are two different topologies available in this quotient space which need not be the same. For instance, in Lemma 7.5, the convergence under the if condition  $[\theta_i] \to [\theta]$  is required to hold in the topology generated by the metric  $\bar{d}$  which is a stronger requirement than merely asking for convergence in the quotient topology. The proofs of the above results are deferred to the relevant subsection in the appendices.
8 Limitations and Broader Impact
In this work, we were primarily focused to establish the consistency of maximum likelihood estimates of the PPCA model in a quotient Euclidean space. However, we were neither able to provide a rate of convergence nor we could say something definitive about the asymptotic distribution of the MLE, and this opens up two avenues for future research. Nonetheless, it is worth noting that our research does not present any discernible negative societal implications.
In this paper, we proposed a novel topological framework to provide rigorous justification for the theoretical validity of the maximum likelihood estimation of the Probabilistic Principal Component Analysis (PPCA) model. Although our consistency results hold within a quotient space, implying that the true parameter is recoverable up to a closed subset of the parameter space, our results establish that this represents the optimal achievable outcome due to the challenges caused by identifiability inherent in the underlying model. Our result is stated in terms of the abstract framework of quotient topological spaces, but an immediate (and concrete) consequence of our work is (strong) consistent covariance estimation of the PPCA process through maximum likelihood estimates, which wasn't known before. Our rigorous framework opens doors for applications in many other statistical models where rotational ambiguity (or more generally ambiguity due to a closed space or symmetric parametrization) is present. One such example is the matrix factorization problem where one is interested in the estimation of two matrices  $\mathbf{A}$ ,  $\mathbf{B}$  such that the data matrix  $\mathbf{X} = \mathbf{AB} + \mathbf{\nabla}$  noise is one such problem. Our methodology is highly flexible in the sense that it does not depend on the statistical model much as long as there are some regularities in the geometry of the parameter space  $\Theta$ , which makes it even more usable, even for non-linear models such as nonlinear independent component analysis where the observed data is generated as  $x = \xi (z|\theta) + \mathrm{noise}$ , where  $z$  is a latent random vector and  $\theta$  is a parameter and the function  $\xi$  is non linear. The methodology we developed in our work for the PPCA model could be readily applied in this case as our quotient space construction barely depends on the fact that  $\xi$  is linear for PPCA. The primary focus for us was to build a strong theoretical foundation which was missing in the relevant literature. Furthermore, our study offers a concise and comprehensive treatment of the theory of quotient topological spaces, with an eye on statistical applications, building upon and expanding upon previous works with enhanced rigour and clarity.
The first author would like to thank his supervisors, Prof. Philippe Gagnon and Prof. Florian Maire, for many helpful conversations and continuous support by providing generous financial assistance and excellent working conditions during this project. The authors would like to express their gratitude to the anonymous reviewers and area chairs of NeurIPS 2023 for their careful review of the paper and their valuable comments and suggestions, which have greatly contributed to improving earlier versions of the manuscript. The authors would also like to thank Prof. Nik Weaver in the mathematics department at Washington University in St. Louis for an insightful private communication. The first author received financial support from his supervisors and fellowships provided by the Faculté des études supérieures et postdoctorales (FESP) and the Bourse d'exemption at the Université de Montréal during the course of this work. The first author is sincerely thankful to his family and remains deeply indebted to his special friend who goes by the nickname 1915131 for providing support of all kinds unconditionally throughout the duration of the work.
The second author would like to acknowledge the financial assistance and excellent working condition provided by Northwestern University.