Fast, Asymptotically Efficient, Recursive Estimation in a Riemannian Manifold

Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a statistical parameter which belongs to a Riemannian manifold. Roughly, this task amounts to stochastic minimisation of a statistical divergence function. The following problem is considered: how to obtain fast, asymptotically efficient, recursive estimates, using a Riemannian stochastic optimisation algorithm with decreasing step sizes. In solving this problem, several original results are introduced. First, without any convexity assumptions on the divergence function, we proved that, with an adequate choice of step sizes, the algorithm computes recursive estimates which achieve a fast non-asymptotic rate of convergence. Second, the asymptotic normality of these recursive estimates is proved by employing a novel linearisation technique. Third, it is proved that, when the Fisher information metric is used to guide the algorithm, these recursive estimates achieve an optimal asymptotic rate of convergence, in the sense that they become asymptotically efficient. These results, while relatively familiar in the Euclidean context, are here formulated and proved for the first time in the Riemannian context. In addition, they are illustrated with a numerical application to the recursive estimation of elliptically contoured distributions.

1. Introduction.Over the last five years, the data science community has devoted significant attention to stochastic optimisation in Riemannian manifolds.This was impulsed by Bonnabel, who proved the convergence of the Riemannian stochastic gradient method [7].Later on [32], the rate of convergence of this method was studied in detail, under various convexity assumptions on the cost function.More recently, asymptotic efficiency of the averaged Riemannian stochastic gradient method was proved in [29].Previously, for the specific problem of computing Riemannian means, several results on the convergence and asymptotic normality of Riemannian stochastic optimisation methods had been obtained [4] [3].
The present work moves in a different direction, focusing on recursive estimation in Riemannian manifolds.While recursive estimation is a special case of stochastic optimisation, it has its own geometric structure, given by the Fisher information metric.Here, several original results will be introduced, which show how this geometric structure can be exploited, to design Riemannian stochastic optimisation algorithms which compute fast, asymptotically efficient, recursive estimates, of a statistical parameter which belongs to a Riemannian manifold.For the first time in the literature, these results extend, from the Euclidean context to the Riemannian context, the classical results of [23] [13].
The mathematical problem, considered in the present work, is formulated in Section 2. This involves a parameterised statistical model P of probability distributions P θ , where the statistical parameter θ belongs to a Riemannian manifold Θ.Given independent observations, with distribution P θ * for some θ * ∈ Θ, the aim is to estimate the unknown parameter θ * .In principle, this is done by minimising a statistical divergence function D(θ), which measures the dissimilarity between P θ and P θ * .Taking advantage of the observations, there are two approaches to minimising D(θ) : stochastic minimisation, which leads to recursive estimation, and empirical minimisation, which leads to classical techniques, such as maximum-likelihood estimation [9] [10].
The original results, obtained in the present work, are stated in Section 3. In particular, these are Propositions 3.2, 3.4, and 3.5.Overall, these propositions show that recursive estimation, which requires less computational resources than maximum-likelihood estimation, can still achieve the same optimal performance, characterised by asymptotic efficiency [18] [31].
To summarise these propositions, consider a sequence of recursive estimates θ n , computed using a Riemannian stochastic optimisation algorithm with decreasing step sizes (n is the number of observations already processed by the algorithm).Informally, under assumptions which guarantee that θ * is an attractive local minimum of D(θ), and that the algorithm is neither too noisy, nor too unstable, in the neighborhood of θ * , • Proposition 3.2 states that, with an adequate choice of step sizes, the θ n achieve a fast non-asymptotic rate of convergence to θ * .Precisely, the expectation of the squared Riemannian distance between θ n and θ * is O (n −1 ).This is called a fast rate, because it is the best achievable, for any step sizes which are O (n −p ) with p ∈ (0, 1) [5] [13].Here, this rate is obtained without any convexity assumptions, for twice differentiable D(θ).It would still hold for non-differentiable, but strongly convex, D(θ) [32].
• Proposition 3.4 states that the distribution of the θ n becomes asymptotically normal, centred at θ * , when n grows increasingly large, and also characterises the corresponding asymptotic covariance matrix.This proposition is proved using a novel linearisation technique, which also plays a central role in [29].
• Proposition 3.5 states that, if the Riemannian manifold Θ is equipped with the Fisher information metric of the statistical model P , then Riemannian gradient descent with respect to this information metric, when used to minimise D(θ), computes recursive estimates θ n which are asymptotically efficient, achieving the optimal asymptotic rate of convergence, given by the Cramér-Rao lower bound.This is illustrated, with a numerical application to the recursive estimation of elliptically contoured distributions, in Section 4.
The end result of Proposition 3.5 is asymptotic efficiency, achieved using the Fisher information metric.In [29], an alternative route to asymptotic efficiency is proposed, using the averaged Riemannian stochastic gradient method.This method does not require any prior knowledge of the Fisher information metric, but has an additional computational cost, which comes from computing on-line Riemannian averages.
The proofs of Propositions 3.2, 3.4, and 3.5, are detailed in Section 5, and Appendices A and B. Necessary background, about the Fisher information metric (in short, this will be called the information metric), is recalled in Appendix C. Before going on, the reader should note that the summation convention of differential geometry is used throughout the following, when working in local coordinates.
2. Problem statement.Let P = (P, Θ, X) be a statistical model, with parameter space Θ and sample space X.To each θ ∈ Θ, the model P associates a probability distribution P θ on X.Here, Θ is a C r Riemannian manifold with r > 3, and X is any measurable space.The Riemannian metric of Θ will be denoted •, • , with its Riemannian distance d(•, •).In general, the metric •, • is not the information metric of the model P .
Let (Ω, F, P) be a complete probability space, and (x n ; n = 1, 2, . ..) be i.i.d.random variables on Ω, with values in X.While the distribution of x n is unknown, it is assumed to belong to the model P .That is, P • x −1 n = P θ * for some θ * ∈ Θ, to be called the true parameter.
Consider the following problem : how to obtain fast, asymptotically efficient, recursive estimates θ n of the true parameter θ * , based on observations of the random variables x n ?The present work proposes to solve this problem through a detailed study of the decreasing-stepsize algorithm, which computes (2.1a) starting from an initial guess θ 0 .This algorithm has three ingredients.First, Exp denotes the Riemannian exponential map of the metric •, • of Θ [26].Second, the step sizes γ n are strictly positive, decreasing, and verify the usual conditions for stochastic approximation [23] [20] (2.1b) ) is a continuous vector field on Θ for each x ∈ X, which generalises the classical concept of score statistic [18] [16].It will become clear, from the results given in Section 3, that the solution of the above-stated problem depends on the choice of each one of these three ingredients.
A priori knowledge about the model P is injected into Algorithm (2.1a) using a divergence function D(θ) = D(P θ * , P θ ).As defined in [2], this is a positive function, equal to zero if and only if P θ = P θ * , and with positive definite Hessian at θ = θ * .Since one expects that minimising D(θ) will lead to estimating θ * , it is natural to require that In other words, that u(θ, x) is an unbiased estimator of minus the Riemannian gradient of D(θ).With u(θ, x) given by (2.1c), Algorithm (2.1a) is a Riemannian stochastic gradient descent, of the form considered in [7][32] [29].However, as explained in Remark 3.7, (2.1c) may be replaced by the weaker condition (3.6), without affecting the results in Section 3. In this sense, Algorithm (2.1a) is more general than Riemannian stochastic gradient descent.
In practice, a suitable choice of D(θ) is often the Kullback-Leibler divergence [27], (2.2a) dP θ * where P θ is absolutely continuous with respect to P θ * with Radon-Nikodym derivative L(θ).Indeed, if D(θ) is chosen to be the Kullback-Leibler divergence, then (2.1c) is satisfied by which, in many practical situations, can be evaluated directly, without any knowledge of θ * .
3. Main results.The motivation of the following Propositions 3.1 to 3.5 is to provide general conditions, which guarantee that Algorithm (2.1a) computes fast, asymptotically efficient, recursive estimates θ n of the true parameter θ * .In the statement of these propositions, it is implicitly assumed that conditions (2.1b) and (2.1c) are verified.Moreover, the following assumptions are considered.
(d1) the divergence function D(θ) has an isolated stationary point at θ = θ * , and Lipschitz gradient in a neighborhood of this point.
(d2) this stationary point is moreover attractive : D(θ) is twice differentiable at θ = θ * , with positive definite Hessian at this point.
(u1) in a neighborhood of θ = θ * , the function For Assumption (d1), the definition of a Lipschitz vector field on a Riemannian manifold may be found in [22].For Assumptions (u1) and (u2), • denotes the Riemannian norm.
Propositions 3.1 to 3.5 require the condition that the recursive estimates θ n are stable, which means that all the θ n lie in Θ * , almost surely.The need for this condition is discussed in Remark 3.8.Note that, if θ n lies in Θ * , then θ n is determined by its normal coordinates θ α n .Proposition 3.1 (consistency).assume (d1) and (u1) are verified, and the recursive estimates θ n are stable.Then, lim θ n = θ * almost surely.Proposition 3.2 (mean-square rate).assume (d1), (d2) and (u1) are verified, the recursive estimates θ n are stable, and γ n = a n where 2λa > 1.
(ii) if a = 1, the distribution of the re-scaled coordinates (n 1/2 θ α n ) converges to a centred dvariate normal distribution, with covariance matrix Σ * .(iii) if a = 1, and u(θ, x) is given by (2.2b), then Σ * is the identity matrix, and the recursive estimates θ n are asymptotically efficient.(iv) the following rates of convergence also hold The following remarks are concerned with the scope of Assumptions (d1), (d2), (u1), and (u2), and with the applicability of Propositions 3.1 to 3.5.
Remark 3.6.(d2), (u1) and (u2) do not depend on the Riemannian metric •, • of Θ. Precisely, if they are verified for one Riemannian metric on Θ, then they are verified for any Riemannian metric on Θ.Moreover, if the function D(θ) is C 2 , then the same is true for (d1).In this case, Propositions 3.1 to 3.5 apply for any Riemannian metric on Θ, so that the choice of the metric •, • is a purely practical matter, to be decided according to applications.Remark 3.7.the conclusion of Proposition 3.1 continues to hold, if (2.1c) is replaced by Then, it is even possible to preserve Propositions 3.2, 3.3, and 3.4, provided (d2) is replaced by the assumption that the mean vector field, X(θ) = E θ * u(θ, x), has an attractive stationary point at θ = θ * .This generalisation of Propositions 3.1 to 3.4 can be achieved following essentially the same approach as laid out in Section 5.However, in the present work, it will not be carried out in detail.
Remark 3.8.the condition that the recursive estimates θ n are stable is standard in all prior work on stochastic optimisation in manifolds [7][32] [29].In practice, this condition can be enforced through replacing Algorithm (2.1a) by a so-called projected or truncated algorithm.This is identical to (2.1a), except that θ n is projected back onto the neighborhood Θ * of θ * , whenever it falls outside of this neighborhood [23] [20].On the other hand, if the θ n are not required to be stable, but (d1) and (u1) are replaced by global assumptions, (d1') D(θ) has compact level sets and globally Lipschitz gradient.
for some constant C and for all θ ∈ Θ.
then, applying the same arguments as in the proof of Proposition 3.1, it follows that the θ n converge to the set of stationary points of D(θ), almost surely.
Remark 3.9.from (ii) and (iii) of Proposition 3.5, it follows that the distribution of n d 2 (θ n , θ * ) converges to a χ 2 -distribution with d degrees of freedom.This provides a practical means of confirming the asymptotic efficiency of the recursive estimates θ n .

Application : estimation of ECD.
Here, the conclusion of Proposition 3.5 is illustrated, by applying Algorithm (2.1a) to the estimation of elliptically contoured distributions (ECD) [14] [28].Precisely, in the notation of Section 2, let Θ = P m the space of m×m positive definite matrices, and X = R m .Moreover, let each P θ have probability density function where h : R → R is fixed, has negative values, and is decreasing, and † denotes the transpose.Then, P θ is called an ECD with scatter matrix θ.To begin, let (x n ; n = 1, 2, . ..) be i.i.d.random vectors in R m , with distribution P θ * given by (4.1), and consider the problem of estimating the true scatter matrix θ * .The standard approach to this problem is based on maximum-likelihood estimation [24][28].An original approach, based on recursive estimation, is now introduced using Algorithm (2.1a).
As in Proposition 3.5, the parameter space P m will be equipped with the information metric of the statistical model P just described.In [6], it is proved that this information metric is an affine-invariant metric on P m .In other words, it is of the general form [21] (4.2a) parameterised by constants I 1 > 0 and I 2 ≥ 0, where tr denotes the trace and tr 2 the squared trace.Precisely [6], for the information metric of the model P , (4.2b) where ϕ is a further constant, given by the expectation 2 with e ∈ P m the identity matrix, and h the derivative of h.This expression of the information metric can now be used to specify Algorithm (2.1a).First, since the information metric is affine-invariant, it is enough to recall that all affineinvariant metrics on P m have the same Riemannian exponential map [25][28], where exp denotes the matrix exponential.Second, as in (ii) of Proposition 3.5, choose the sequence of step sizes (4.3b) Third, as in (iii) of Proposition 3.5, let u(θ, x) be the vector field on P m given by (2.2b), where ∇ (inf ) denotes the gradient with respect to the information metric, and L(θ) is the likelihood ratio, equal to p(x|θ) divided by p(x|θ * ).Now, replacing (4.3) into (2.1a)defines an original algorithm for recursive estimation of the true scatter matrix θ * .
To apply this algorithm in practice, one may evaluate u(θ, x) via the following steps.Denote g(θ, x) the gradient of log p(x|θ) with respect to the affine-invariant metric of [25], which corresponds to I 1 = 1 and I 2 = 0.By direct calculation from (4.1), this is given by Moreover, introduce the constants J 1 = I 1 and J 2 = I 1 + mI 2 .Then, u(θ, x) can be evaluated, from the orthogonal decomposition of g = g(θ, x),  5. Proofs of main results.
5.1.Proof of Proposition 3.1.the proof is a generalisation of the original proof in [7], itself modeled on the proof for the Euclidean case in [8].Throughout the following, let X n be the σ-field generated by x 1 , . . ., x n [27].Recall that (x n ; n = 1, 2, . ..) are i.i.d. with distribution P θ * .Therefore, by (2.1a), θ n is X n -measurable and x n+1 is independent from X n .Thus, using elementary properties of conditional expectation [27], where (5.1a) follows from (2.1c), and (5.1b) from (u1).Let L be a Lipschitz constant for ∇D(θ), and C be an upper bound on V (θ), for θ ∈ Θ * .The following inequality is now proved, for any positive integer n, once this is done, Proposition 3.1 is obtained by applying the Robbins-Siegmund theorem [13].
Proof of (5.2) : let c(t) be the geodesic connecting θ n to θ n+1 with equation From the fundamental theorem of calculus, Since the recursive estimates θ n are stable, θ n and θ n+1 both lie in Θ * .Since Θ * is convex, the whole geodesic c(t) lies in Θ * .Then, since ∇D(θ) is Lipschitz on Θ * , it follows from (5.3b), (5.3c) Taking conditional expectations in this inequality, and using (5.1a) and (5.1b), (5.3d) Conclusion : by the Robbins-Siegmund theorem, inequality (5.2) implies that, almost surely, In particular, from the first condition in (2.1b), convergence of the sum in (5.4a) implies 5.2.Proof of Proposition 3.2.the proof is modeled on the proofs for the Euclidean case, given in [23] [5].It relies on the following geometric Lemmas 5.1 and 5.2.Lemma 5.1 will be proved in Appendix A. On the other hand, Lemma 5.2 is the same as the trigonometric distance bound of [32].For Lemma 5.1, recall that λ > 0 denotes the smallest eigenvalue of the matrix H defined in (3.1b).
Proof of (3.2) : let γ n = a n with 2λa > 2µa > 1 for some µ < λ, and let Θ * be the neighborhood corresponding to µ in Lemma 5.1.By Proposition 3.1, the θ n converge to θ * almost surely.Without loss of generality, it can be assumed that all the θ n lie in Θ * , almost surely.Then, (2.1a) and Lemma 5.2 imply, for any positive integer n, Indeed, this follows by replacing τ = θ n+1 and θ = θ n in (5.5b).Taking conditional expectations in (5.6a), and using (5.1a) and (5.1b), Then, by (u1) and (5.5a) of Lemma 5.1, where C is an upper bound on V (θ), for θ ∈ Θ * .By further taking expectations Using (5.6c), the proof reduces to an elementary reasoning by recurrence.Indeed, replacing γ n = a n into (5.6c), it follows that Let b be sufficiently large, so (5.7b) is verified and E d 2 (θ no , θ * ) ≤ b(n o ) for some n o .Then, by recurrence, using (5.7a) and (5.7b), one also has that In other words, (3.2) holds true.
5.3.Proof of Proposition 3.3.the proof is modeled on the proof for the Euclidean case in [23].To begin, let W n be the stochastic process given by (5.8a) W n = n p d 2 (θ n , θ * ) + n −q where q ∈ (0, 1 − p) The idea is to show that this process is a positive supermartingale, for sufficiently large n.By the supermartingale convergence theorem [27], it then follows that W n converges to a finite limit, almost surely.In particular, this implies Then, p must be equal to zero, since p is arbitrary in the interval (0, 1).Precisely, for any ε ∈ (0, 1 − p), a supermartingale, for sufficiently large n.To do so, note that by (5.6b) from the proof of Proposition 3.2, Here, the first term on the right-hand side is negative, since 2µa > 1 > p.Moreover, the third term dominates the second one for sufficiently large n, since q < 1 − p.Thus, for sufficiently large n, the right-hand side is negative, and W n is a supermartingale.Proposition 3.4.the proof relies on the following geometric Lemmas 5.3 and 5.4, which are used to linearise Algorithm (2.1a), in terms of the normal coordinates θ α .This idea of linearisation in terms of local coordinates also plays a central role in [29].
Study of equation (5.10b) : switching to vector-matrix notation, equation (5.10b) is of the general form (5.11a) where I denotes the identity matrix, A has matrix elements A αβ , and (ξ n ) is a sequence of inputs.The general solution of this equation is [23] [19] (5.11b) where the transition matrix A n,k is given by (5.11c) Since 2λa > 1, the matrix A is stable.This can be used to show that [23][19] (5.11d) q > 1 2 and E |ξ n | = O(n −q ) =⇒ lim η n = 0 in probability where |ξ n | denotes the Euclidean vector norm.Then, it follows from (5.11d) that η n converges to zero in probability, in each one of the three cases n + 1 Indeed, in the first two cases, the condition required in (5.11d) can be verified using (3.2), whereas in the third case, it follows immediately from the estimate of E|π α n+1 | in (5.9a).Conclusion : by linearity of (5.10b), it is enough to consider the case ξ α n+1 = w α n+1 in (5.11a).Then, according to (5.11b), η n has the same limit distribution as the sums (5.12) ηn = By (5.1), (w k ) is a sequence of square-integrable martingale differences.Therefore, to conclude that the limit distribution of ηn is a centred d-variate normal distribution, with covariance matrix Σ given by (3.4), it is enough to verify the conditions of the martingale central limit theorem [15], (5.13a) lim max where Σ k is the conditional covariance matrix (5.13) are verified in Appendix B, which completes the proof.5.5.Proof of Proposition 3.5.denote ∂ α = ∂ ∂θ α the coordinate vector fields of the normal coordinates θ α .Since •, • coincides with the information metric of the model P , it follows from (3.1b) and (C.1), (5.15a) However, by the definition of normal coordinates [26], the ∂ α are orthonormal at θ * .Therefore, (5.15b) Thus, the matrix H is equal to the identity matrix, and its smallest eigenvalue is λ = 1.
For the following proof of (iii), the reader may wish to recall that summation convention is used throughout the present work.That is [26], summation is implicitly understood over any repeated subscript or superscript from the Greek alphabet, taking the values 1 , . . ., d .
Proof of (iii) : let (θ) = log L(θ) and assume u(θ, x) is given by (2.2b).Then, by the definition of normal coordinates [26], the following expression holds (5.16a) Replacing this into (3.1a)gives where the second equality is the so-called Fisher's identity (see [2], Page 28), and the third equality follows from (2.2a) by differentiating under the expectation.Now, by (3.1b) and (5.15b), Σ * is the identity matrix.
To show that the recursive estimates θ n are asymptotically efficient, let (τ α ; α = 1, . . ., d ) be any local coordinates with origin at θ * and let τ α n = τ α (θ n ) .From the second-order Taylor expansion of each coordinate function τ α , it is straightforward to show that (5.17a) where the subscript θ * indicates the derivative is evaluated at θ * , and where σ α is a continuous function in the neighborhood of θ * .By (3.3), the second term in (5.17a) converges to zero almost surely.Therefore, the limit distribution of the re-scaled coordinates (n 1/2 τ α n ) is the same as that of the first term in (5.17a).By (ii), this is a centred d-variate normal distribution with covariance matrix Σ τ given by where the second equality follows because Σ * γκ = δ γκ since Σ * is the identity matrix.It remains to show that Σ τ is the inverse of the information matrix I τ as in (C.3).According to (C.1), this is given by (5.17c) where the second equality follows from (2.2a), and the third equality from Fisher's identity (see [2], Page 28).Now, a direct application of the chain rule yields the following By the first equality in (5.16b), this is equal to (5.17d) because Σ * γκ = δ γκ is the identity matrix.Comparing (5.17b) to (5.17d), it is clear that Σ τ is the inverse of the information matrix I τ as in (C.3).
Condition (5.13b) : to verify this condition, recall that (w k ) is a sequence of square-integrable martingale differences.Therefore, from (5.12) where Σ k is the conditional covariance matrix in (5.14).Applying (B.1) to each term under the sum in (B.3a), it follows that where d is the dimension of the parameter space Θ, and |Σ k | F denotes the Frobenius matrix norm.However, it follows from (u1) that there exists a uniform upper bound S on |Σ k | F .Therefore, by (B.3b) Since ν > 0, the right-hand side of (B.3c) remains bounded as n → ∞, by the Euler-Maclaurin formula [12].This immediately yields Condition (5.13b).
Condition (5.13c) : to verify this condition, it is first admitted that the following limit is known to hold where Σ * was defined in (3.1a).Then, let the sum in (5.13c) be written Due to the equivalence A n,k ∼ exp(ln(n/k)A) (see [23], Page 125), the first term in (B.4b) is a Riemann sum for the integral [23][19] which is known to be the solution Σ of Lyapunov's equation (3.4).The second term in (B.4b) can be shown to converge to zero in probability, using inequality (B.1) and the limit (B.4a), by a similar argument to the ones in the verification of Conditions (5.13a) and (5.13b).Then, Condition (5.13c) follows immediately.
Proof of (B.4a) : recall that Since (w k ) is a sequence of square-integrable martingale differences, it is possible to write, in the notation of (5.14), By (5.4b), the second term in (B.5a) converges to zero almost surely, as k → ∞.It also converges to zero in expectation, since ∇D(θ) is uniformly bounded for θ in the compact set Θ * .For the first term in (B.5a), since the x k are i.i.d. with distribution P θ * , it follows that Since u(θ, x) is a continuous vector field on Θ for each x ∈ X, and θ k−1 converge to θ * almost surely, it follows that u(θ k−1 , x) converge to u(θ * , x) for each x ∈ X, almost surely.On the other hand, it follows from (u2) that the functions under the expectation E θ * in (B.5b) have bounded second order moments, so they are uniformly integrable [27].Therefore, In any change of coordinates, these transform like the components of a (0, 2) covariant tensor [26].That is, if (θ α ; α = 1, . . .where the subscript θ * indicates the derivative is evaluated at θ * , and where I θ γκ are the components of the information matrix I θ of the coordinates θ α .
The recursive estimates θ n are said to be asymptotically efficient, if they are asymptotically efficient in any local coordinates τ α , with origin at θ * .That is, according to the classical definition of asymptotic efficiency [18][31], if the following weak limit of probability distributions is verified [27], where L{. ..} denotes the probability distribution of the quantity in braces, τ α n = τ α (θ n ) are the coordinates of the recursive estimates θ n , and N d (0, Σ τ ) denotes a centred d-variate normal distribution with covariance matrix Σ τ .
It is important to note that asymptotic efficiency of the recursive estimates θ n is an intrinsic geometric property, which does not depend on the particular choice of local coordinates τ α , with origin at θ * .This can be seen from the transformation rule of the components of the information matrix, described above.In fact, since these transform like the components of a (0, 2) covariant tensor, the components of Σ τ transform like those of a (2, 0) contravariant tensor, which is the correct transformation rule for the components of a covariance matrix.

Figures 1
Figures 1 and 2 below display numerical results from an application to Kotz-type distributions, which correspond to h(t) = − t s 2 in (4.1) and ϕ = s 2 m 2s
ν where |A n,k |