Quantum Expectation-Maximization for Gaussian Mixture Models
Authors: Iordanis Kerenidis
Anupam Prakash
Alessandro Luongo
Venue:  (2020)
Keywords: Quantum
ExpectationMaximization
Unsupervised
QRAM

Abstract: The Expectation-Maximization (EM) algorithm is a fundamental tool in unsupervised machine learning. It is often used as an efficient way to solve Maximum Likelihood (ML) and Maximum A Posteriori estimation problems, especially for models with latent variables. It is also the algorithm of choice to fit mixture models: generative models that represent unlabelled points originating from $k$ different processes, as samples from $k$ multivariate distributions. In this work we define and use a quantum version of EM to fit a Gaussian Mixture Model. Given quantum access to a dataset of $n$ vectors of dimension $d$, our algorithm has convergence and precision guarantees similar to the classical algorithm, but the runtime is only polylogarithmic in the number of elements in the training set, and is polynomial in other parameters - as the dimension of the feature space, and the number of components in the mixture. We generalize further the algorithm by fitting any mixture model of base distributions in the exponential family. We discuss the performance of the algorithm on datasets that are expected to be classified successfully by those algorithms, arguing that on those cases we can give strong guarantees on the runtime.

QUANTUM EXPECTATION MAXIMIZATION FOR GAUSSIAN MIX- TURE MODELS
Anonymous authors Paper under double-blind review
ABSTRACT
The Expectation- pnpcctaionnMaxxmizzttoo (EM) algorithm is a fundamental tool in unsupervised machine learning. is often used as an efficient way to solve Maximum Likelihood (ML) and Maximum A Posteriori estimation problems, especially for models with latent variables. It is also the algo- rithm of choice to fit mixture models: generative models that represent unlabelled points originating from & different processes, as samples from k multivariate distributions. In this work we define and use a quantum version of EM to fit Gaussian Mixture Model. Given quantum access to a dataset of n vectors of dimension d, our algorithm has convergence and precision guarantees similar to the classical algorithm, but the runtime is only polylogarithmic in the number of elements in the training set, and is polynomial in other parameters as the dimension of the feature space, and the number of components in the mixture. We generalize further the algorithm by fitting any mixture model of base distributions in the exponential family. We discuss the performance of the algorithm on datasets that are expected to be classified successfully by those algorithms, arguing that on those cases we can give strong guarantees on the runtime.
1 INTRODUCTION
Over the last few years, the effort to find real world applications of quantum computers has greatly intensified. Along with chemistry, material sciences, finance, one of the fields where quantum computers are expected to be most beneficial is machine learning. A number of different algorithms have been proposed for quantum machine learning
(Biamonte et al., 2017: Wiebe et al., 2017; Kerenidis & Prakash 2018 Harrow et al.] 2009; Subaşi et al.] 2019; Farhi & Neven, 2018), both for the supervised and unsupervised setting, and despite the lack of large-scale quantum computers and quantum memory devises, some quantum algorithms have been demonstrated in proof-of- pfriincipeee experiments (Li al., 2015 Otterbach al. 2017: Jiang et al. 2019). Here, we look
appectatio-MMaxinzzatnn (EM), a fundamental algorithm in unsupervised learning, that can be used to fit different mixture models and give maximum likelihood estimates with the so-called latent variable models. Such generative models are one of the most promising approaches for unsupervised problems. The goal of generative model is to learn a probability distribution that is most likely to have generated the data collected in a training set V E Ruxd of n vectors of d features. Fitting the model consists in learning the parameters of probability distribution in a certain parameterized family that best describes our vectors v,. We will see that, thanks to this formulation, we can reduce a statistical problem into an optimization problem using maximum likelihood estimation (ML) estimation. The likelihood is the function that we use to measure how good a model is for explaining given dataset. For a given machine learning model with parameters 7, the likelihood of our data set V is the probability that the data have been generated by the model with parameters 7, assuming each point is independent and identically distributed. We think the likelihood as function of 7, holding the dataset V fixed. For p(u./7) the probability that point va comes from model 7, the likelihood is defined as T.(;;; V) = I (())))). From this formula, we can see that in order to find the best parameters > of our model we need to solve an optimization problem. For numerical and analytical reasons, instead of maximizing the likelihood L, it is common practice to find the best model by maximizing the log-likelihood function E(7; V) = log L(T; V) " log ((g/())))) In this context, we want to find the model that maximizes the log-likelioood: 7ML arg max.,
A log p(v)))) The procedure to calculate the log-likelihood depends the specific model under consideration. A possible solution would be to use a gradient based optimization algorithm on l. Unfortunately, due to the indented landscape of the function, gradient based techniques often do not perform well. Therefore, it is common to solve the maximum likelihood estimation (or maximum a priori) problem using the Expectation-
Cotettatiio-Maxiizzttttt Maximization (EM) algorithm. EM is an iterative algorithm which is guaranteed converge to a (local)

---

optimum of the likelihood. This algorithm has striking variety of applications, and has been successfully used for medical imaging (Balafar et al., 2010), image restoration (Lagendijk et al., 1990), problems in computational biology (Fan et al. 2010), and so on. EM has been proposed in different works by different authors, but has been formalized as we know it only in 1977 (Dempster et al. 1977). For more details, we refer to (Lindsay 1995; Bilmes et al. 1998).
In this work, we introduce Quantum Dnpectaaion-Maximizzane We detail its usage in the context of Gaussian Mixture Models, and we extend the result to other distributions in the exponential family. We also generalize the result by showing how to compute the MAP: the Maximum A Posteriori estimate of mixture model. MAP estimates can be seen as the Bayesian version of maximum likelihood estimation problems. MAP estimates are often preferred over MI. estimates, due to a reduced propensity to overfit. Our main result can be stated as:
(QEM), a new algorithm for fitting mixture models,
Result (Quantum Expectation- Expectation-Maximizaaa Maxximzzaiiiii)... (see Theorem 3.9 For a data matrix V E Rnxd stored in appropri- ate QRAM data structure and for parameters ōg Sp > O, Quantum Expectation- Maximization (QEM) fits a Maximum Likelihood (or Maximum A Posteriori) estimate of Gaussian Mixture Model with k components, in running time per iteration which is dominated by:
n R2 1 (V))
(1)
84
where E' is a covariance matrix of a Gaussian distribution, 17 is parameter of the dataset related to the maximum norm of the vectors, da, 8,. are error parameters in the QEM algorithm, uu< vd) is a factor appearing in quantum linear algebra and k is the condition number of a matrix. Here we only kept the term in the running time that dominates for the range of parameters of interest. In Theorem 3.9 we explicate the rumning time of cach step of the algorithm. The QEM algorithm runs for a number of iterations until a stopping condition is met (defined by a parameter 6, > 0) which implies convergence to a (local) optimum.
Let's have a first high-level comparison of this result with the standard classical algorithms. The runtime of a single iteration in the standard implementation of the EM algorithm is at least Okndd) (Pedregosa et al., 2011 Murphy 2012). The advantage of the quantum algorithm is an exponential improvement with respect to the number of elements in the training set, albeit with worsening on other parameters. It is crucial to find datasets where such a quantum
algorithm can offer speedup. For a reasonable range of parameters (d = 40, k = 10, r = 10, δ = 0.5, k(V)- = 25, x(E)= 5, p(Z)= 4) which is motivated by some experimental evidence reported in Section datasets where the number of samples in the order of O(1012) might be processed faster on quantum computer. One should expect that some of the parameters of the quantum algorithm can be improved, especially the dependence on the condition numbers and the errors, which can make enlarge the type of datasets where QEM can offer an advantage. Note that we expect the number of iterations of the quantum algorithm to be proportional to the number of iteration of the classical case. This is to be expected since the convergence rate does not change, and it is corroborated by previous experimental evidence in a similar scenario: the number of iterations needed by q-means algorithm for convergence. is proportional to the number of iterations of the classical k-means algorithm (Kerenidis et al. 2018).
Expectation- Matiatiioo--aiiizittiionn is widely used for fitting mixture models in machine learning (Mrrphy 2012). Most mix- ture models use base distribution in the exponential family: Poisson (Church & Gale 1995), Binomial, Multinomial, log-normal (Dexter & Tanner 1972). exponential (Ghitany et al.) 1994). Dirichlet multinomial (Yin & Wang 2014). and others. EM is also used to fit mixtures of experts, mixtures of the student T distribution (which does not belong to the exponential family, and can be fitted with EM using (Liu & Rubin 1995)) and for factor analysis, probit regression, and learning Hidden Markov Models (Murphy 2012).
1.1 PREVIOUS WORK
There is fair amount of quantum algorithms that have been proposed in the context of unsupervised learning. (Aimeur al. 2013) Lloyd et al. 2013b] Otterbach et al. 2017). Recently, classical machine learning algorithms were obtained by "dequantizing" quantum machine learning algorithms (Tang) 2018b | Gilyén al. 20188a] Chia al.) 2018). The runtime of these classical algorithm is poly-logarithmic in the dimensions of the dataset. However, the polynomial dependence on the rank, the error, and the condition number, make these new algorithms impractical on interesting datasets, as shown experimentally by (Arrazola et al. 2019). Fast classical algorithm for GMM exists, albeit assuming only one shared covariance matrix (Dasgupta) 1999). and without a polylogarithmic dependence in the number of
2

---

elements in the training set. Independently of us, Miyahara, Aihara. and Lechner extended the q-means algorithm (Kerenidis et al. 2018) for Gaussian Mixture Models (Miyahara et al. 2019), using similar techniques. The main difference is that in their work the update step is performed using a hard-clustering approach (as in the k-means algorithm), that is for updating the centroid and the covariance matrices of a cluster j. only the data points for which cluster j is nearest are taken into account. In our work, we use the soft clustering approach (as in the classical EM algorithm), that is for updating the centroid and the covariance matrices of cluster j. all the data points weighted by their responsibility for cluster j are taken into account. Both approaches have merits and can offer advantages (Kearns) et al. 1998).
2 EXPECTATION--AXXIIIZTTII AND GAUSSIAN MIXTURE MODELS
As is common in machine learning literature, we introduce the Expectation- -atitttion-Mimmizzttioon algorithm by using it to fit Gaussian Mixture Models (GMM). Mixture models are a popular generative model in machine learning. The intuition behind mixture models is to model complicated distributions by using group of simpler (usually uni-modal) distri- butions. In this setting, the purpose of the learner is to model the data by fitting the joint probability distribution which most likely have generated our samples. In this section we describe GMM: probably the most used mixture model used to solve unsupervised classification problems. In fact, given a sufficiently large number of mixture components, it is possible to approximate any density defined in Rd (Murphy 2012). In unsupervised settings, we are given training set of unlabeled vectors U1 Un E Rd which we represent as rows of a matrix V E R=xd Let y E [k] one of the k possible labels for a point Ug. We posit that the joint probability distribution of the data p(u,, y4) = p(e, (()))) is defined as follow: Ve ~ Multinomial(() for 0 E TR* and p( (ily j) ~ N(,, Σj). The Og are the mixing weights, i.e. the probabilities that yi = j. and (ugg Σj) is the Gaussian distribution centered in M4 E Rdd with covariance matrix Σn E Raxd, Note that the variables ys are unobserved, and thus are called latent variables. There is a simple interpretation for this model. We assume the data created by first selecting an index j E [k]] by sampling according to Multinomial()), and then a vector v; is sampled from N E,)). Fitting GMM to a dataset reduces to finding an assignment for the parameters: 7 (0. fr, Σ) = (0, j1,... : (Kk En, .... Σx) that best maximize the ligliiol for a given dataset. Note that the algorithm used to fit GMM can return local minimum which might be different than y": the model that represents the global optimum of the likelihood function. We use the letter e to represent our base distribution, which in this case is the probability density function of a multivariate Gaussian distribution (H, E). Using this formulation, a GMM is expressed as: p(u)= = Σド 0, P.j F.y) where 0, are the mixing weights of the multinomial distribution such that Σド= Og 1. The probability for an observation Us to be assigned to the component
A 8y0(maps x2,) j is given by: Fiy = Σm (v, Ex)) This value called responsibility. and corresponds to the posterior probability of the sample i being assigned label j by the current model. As anticipated, to find the best parameters of our gener- ative model, we maximize the log-likelihood of the data. For GMM, the likelihood is given by the following formula Alas, it is seldom possible to solve maximum
(Ng. 20y V)= E(O, ne, E; V) Σ log p(<; : 0, μr, Σ) likelihood estimation analytically (i.e. by finding the zeroes of the derivatives of the log-like function, and this is one of those cases. Fortunately,
DnpettaioonMaxxmizatiioo is an iterative algorithm that solves numerically the optimiza- tion problem of ML estimation. To complicate things, the likelihood function for GMM is not convex, and thus we might find some local minima (Hastie et al. 2009). If we were to know the latent variable ye, then the ligeliieldoodd for GMM would be:f E(>;; V)- -- log p(v: ys 14, Σ) + log P(v;; 0) This formula can be easily maximized with respect to the parameters 8, μt, and Σ. In the Expectation step we calculate the missing variables y 's, given a guess of the parameters (0, μ. E) of the model. Then, in the Maximization step, we use the estimate of the latent variables obtained in the Expectation step to update the estimate of the parameters. While in the Expectation step we calculate a lower bound on the likelihood, in the Maximization step we maximize it. Since at each iteration the likelihood can only increase, the algorithm is guaranteed to converge, albeit possibly to a local optimum (see (Hastic et al. 2009) for the proof). During the Expectation step all the responsibilities are calculated, while in the Maximization step we update our estimate on the parameters 7--1 = (0t+1, p +1) The stopping criterion for GMM is usually a threshold the increment of the log- likelihooh::: if the log- ligellikeihodd changes less than threshold between two iterations, then the algorithm stops. Notice that, since the value of the log-likelihood significantly depends on the amount of data adopt a scale-free stopping criterion, which does not depend on the number of samples. For instance, in the toolkit scikit-learn (Pedregosa et al., 2011) the stopping criterion is given by a tolerance on the average increment of the log-probability, which is chosen to be smaller than certain e,, say 10-3, 3. More precisely, the stopping criterion is [E[log p(u; >*) - Ellog p(v; *f1)]] < e, which we can estimate as E" E log (ogp(u;n") n Ei= log p(v;: ++1) (11))=11))))))))) < 6,.
points in the training sets, it is often preferable
m
3

---

Dataset assumptions in GMM As in q-means Kerenidis et al., 2018), we have an assumption on the dataset that elements of the mixture contribute proportionally to the total responsibility: i, e
vi,le (1)) [k]
EL TM
This is equivalent to requiring that clusters share a comparable amount of points in the "well-cuusterability" assumption in q-means (Kerenidis et al. 2018). It is also equivalent to assuming that 0//0; (1)) Vj, E [A.. For convenience, in this work, we also assume that the dataset is normalized such that the shortest vector has norm and define == mazi ||xl]l to be the maximum norm squared of a vector in the dataset. This is not necessary requirement for our dataset, but it will simplify the analysis of our algorithm, allowing us give strict bounds on the runtime.
Preliminaries We assume a basic understanding of quantum computing, we recommend Nielsen and Chuang (Nielsnn Chuang 2002) for an introduction to the subject. A vector state |o} for E Rd is defined as 2) ||ll| 2 "y (5), where (j) represents Cj, the jth vector in the standard basis. The dataset is represented by a matrix V E Raxd, i.e. each row is a vector &s E Rd for i E [n] that represents a single data point. The cluster centers, called centroids, at time i are stored in the matrix C' E R such that the jth row oj for [k] represents the centroid of the cluster Cj.- We denote as V2, the matrix E ".u.""" where de is the smallest singular value which is greater than With mnz(V) we mean the number of non-zero elements of the rows of V. When we say <(V) we mean the condition number of the matrix V. that the ratio between the biggest and the smallest (non-zero) singular value. All the tools used in this work, like quantum algorithms for computing distances and linear algebraic operations, are reported in the Supplementary Material section.
3 QUANTUM EXPECTATION-MAXIIIAAATI
FOR GMM
In this section, we present quantum Expectation- Expectation-Maximizzatie Maximization algorithm to fit a GMM. The algorithm can also be adapted fit other mixtures models where the probability distributions belong to the exponential family. As the GMM is both intuitive and one of the most widely used mixture models, our results are presented for the GMM case.
A robust version of the EM algorithm Similar to the work of (Kerenidis et al., 2018). we define A A-robust version of the EM algorithm which we use to fit a GMM. The difference between this formalization and the original EM for GMM is simple. Here we explain the numerical error introduced in the training algorithm.
Let 7f = -0', E') = (#', 14= ... .. Ej ... D a model fitted by the standard EM algorithm from 700 an initial guess of the parameters, i.e. 72t is the error-free model that standard EM would have returned after iterations. Starting from the same choice of initial parameters 70, fitting a GMM with the QEM algorithm with A = (6g. 8,,)
means returning a model 3" = (7' F E') such that: j g¹ < 58. that ||zs" - rIll < 8μ for all j E [R:]. and that
Σ - 2 S 8.
Quantum access to the mixture model As in the classical algorithm, we use some subroutines to compute the responsibilities and update our current guess of the parameters. The classical algorithm has clearly two separate steps for Expectation and Maximization. In contrast, the quantum algorithm uses a subroutine to compute the responsibilities inside the step that performs the Maximization, that is the subroutines for computing responsibilities are called multiple times during the quantum Maximization step. During the quantum Maximization step, the algorithm updates the model parameters y by creating quantum states corresponding to parameters y and then recovering classical estimates for these parameters using quantum tomography or amplitude amplification. In order for this subroutines to be efficient, the values of the GMM stored in QRAM data structures and are updated following each maximization step.
Definition 1 (Quantum access to a GMM). We say that we have quantum access to a GMM if the dataset V e R and model parameters O, E R. μ1 E Rdd E, E Rdxd for all j E [k] are stored in QRAM data structures which allow us to perform in time (polylog(d) the following mappings:
bj) 10) + ( (p.) for all j E [A].
1b) (b) J0)
lb> (0) | for j [k], i [d] where of is the i-th rows 2i gaxd
4

---

(t) |0) + {2) le for all i [m].
b>) (0) (0) { (ec[oeef)) (0) (uy) (a)) for all i E [n].
bi> (0) < 102).
Algorithm Quantum Expectation Maximization for GMM
Require: Quantum access to a GMM model, precision parameters 6g, وμ. and threshold c,. Ensure: A GMM 7f that maximizes locally the likelihood f(7; V), up to tolerance 67-
1: Use heuristic described at the beginning of this section to determine initial guess for for = (00, 10,,10 ) and store these parameters in the QRAM. Use Lemma 3.1 to estimate the log determinant of the matrices (20)
t=0
repeat
5: Step 1: Get an estimate of (++1 such that | 0t+1 S 0g using Lemma 3.4
6: Step 2: Get an estimate (155+1)1 Jj=1 by using Lemma 3.6 to estimate each
[1j and (u) such that ++1
|| 1 1 S 6,.
7: Step 3: Get an estimate
1 by using Lemma 3.7 to estimate || and (2)-)) such that
1 - - | S G.
Step 4: Estimate 7 up to error e, / using Theorem 3.8
Step 5: Store e*11 in the QRAM, and Lemma 3.1 estimate the determinants {log det(2 {
10: t=t+1
11:
until
E[p(un>*)] - Elp(unn-' <
12: Return 7" = (0', / )
Quantum initialization strategies exists, and are described in the Appendix.
3.1 EXPECTATION
In this step of the quantum algorithm just showing how to compute efficiently the responsibilities quantum state. First, we compute the responsibilities a quantum register, and then we show how to put them as amplitudes of quantum state. We start by classical algorithm used to efficiently approximate the log-determinant of the covariance matrices of the data. At each iteration Quantum of the updated covariance matrices, which is done thanks to Lemma [.1 We will see from the eтror analysis that in order to get an estimate of the GMM, we need to call Lemma 3.1 with precision for which the runtime of Lemma 3.1 gets subsumed by the running time of finding the updated covariance matrices through 3.7 Thus, do not explicitly write the time to compute the determinant from now on in the algorithm and when we say that we update Σ we include an update on the estimate of (og(de((E)) as well.
Maximization we need to compute the determinant
Lemma 3.1 (Determinant evaluation). There is algorithm that, given as imput matrix E and parameter S with probability 1 1-3 ô in time:
I, outputs an estimate log(det (de(())) such that (
TDete = ((0][)))((()))))n((())))))
Now we can state the main brick used to compute the responsability; a quantum algorithm for evaluating the exponent of a Gaussian distribution.
Lemma 3.2 (Quantum Gaussian Evaluation), Suppose we have stored in the QRAM matrix _gaxd, the centroid E a2d and the covariance matrix Σ E Rdddd of multivariate Gaussian distribution ocvlu, E) as well as an estimate for log)) Then for E O, there exists a quantum algorithm that with probability 1 performs the mapping.
5

---

Ua..: : (0) - l (i) such that |s; - < é1, where s; = - -2 ~ 'ye (4) + d d 2n +
lo)( is the exponent for the Gaussian probability density function.
The running time of the algorithm Ta, =0 (=" (E)))))) 1oR(1/1)))) 1)
c1
Using controlled operations it is simple to extend the previous Theorem to work with multiple Gaussians distributions (pz. Ez). That is, we can control on register (j) to do (5) (i) [0} + (5) (a) [o(eilı;,E)) In the next Lemma we will see how to obtain the responsibilities Túj using the previous Theorem and standard quantum circuits for doing arithmetic, controlled rotations, and amplitude amplification. The Lemma is stated in general way, to be used with any probability distributions that belong to an exponential family.
Lemma 3.3 (Calculating responsibilities). Suppose we have quantum access to GMM with parameters Y (0', p², ). There are quanium algorithus that can:
1. Perform the mapping (8) (5) [0) > (2) ()) (ras) such that [naj - risl S 61 with probability - in time:
TBy.. = =Ō(,. x To,a)
For a given j E [b], construct state R2) such that RS
Σ 1=0
rig 1i)
< E1 where 43 = Σ r with
2
1=0
high probability in time:
TRa = Ō(h² x TRI,e))
3.2 MAXIMIZATION
Now we need to get a new estimate for the parameters of our model. This is the idea: at each iteration we recover the new parameters of the model as quantum states, and then recover it using tomography, amplitude estimation, or sampling. Once the new model has been recovered, we update the QRAM such that we get quantum access to the model 7 2+11_ The possibility to estimate 0 comes from a call to the unitary we built to compute the responsibilities, and postselection.
Lemma 3.4 (Computing f1+1). We assume quantum access to GMM with parameters 72t and let 80 0 be a
precision parameter. There exists algorithun that estimates 0 R& such that |o - -0++1 I ōg in time
To 0 1, (E)))
We use quantum linear algebra to transform the uniform superposition of responsibilities of the j-th mixture into the new centroid of the j-th Gaussian. Let Rj E R" be the vector of responsibilities for a Gaussian j at iteration t. The following claim relates the vectors Rt to the centroids +
Claim 3.5. Let 17, Rn be the vector of responsibilities of the points for the Gaussian j at time i.e. (Rt)s = rls
Then 1+1 En Dtyr
V
Ei no,
The proof is straightforward.
Lemma 3.6 (Computing 11,t),, We assume have quantum access to a GMM with parameters 7². For pre- cision parameter 8, > O, there is quantum algorithm that calculates {uytt1y
+11k 1= such that for all [M]]
| + < 6, in time Tr =Ō [kdne(V)[aV))+ 5 (E)n(E)
63
From the ability to calculate responsibility and indexing the centroids, we derive the ability to reconstruct the covari- ance matrix of the Gaussians well. Again, an approximation of each Σ,. Recall that we have defined the matrix V E Ra where the i-th row of V is defined as vec[u.o?") J. For this Lemma, we assume have the matrix stored in the QRAM. This is a reasonable assumption as the quantum states corresponding to the rows of V' can be prepared as (i) (0) JO) + (t) (u.) [vi), using twice the procedure for creating the rows of V.
quantum linear algebra subroutines and tomography recover

---

Lemma 3.7 (Computing ). ). We assume we have quantuam access to a GMM with parameters x². We also have computed estimates J1 ++1 of all centroids such that - I 5p for precision parameter 6n > Then, there exists a quantum algorithm that outputs estimates for the new covariance matrices {2, 1 such that | E - S Su with high probability, in time,
Ty 0( (M + + (x))((2))
83
3.3 QUANTUM ESTIMATION OF LOG-LIKELIHOOD
Now we are going to show how it is possible get an estimate of the log-likelihood using a quantum procedure and access to GMM model. A good estimate is crucial, as it is used as stopping criteria for the quantum algorithm as well. Classically, we stop to iterate the EM algorithm when 18(7"; V)
=6 +, V)I < ne, or equivalently, we can EE[ p( P(log (( < e. In the quantum algorithm it is more practical estimate ((p(e;)))) = Ei=1 p(u;7) 7)- From this we can estimate an upper bound on the log-likelihode as n log Elp(e;)] = >. log Elp(())) 2 E. logp(p;))) = ((>)
set a tolerance on the average increase in log probability: [E [loogg(m
Lemma 3.8 (Quantum estimation of likelihood). We assume have quantum access GMM with parameters x
For e > 0, there exists quantum algorithm that estimates ([[((n 7)] with absolute error E, in time
Te Ō 1,..,,15 1.5 R² (
(E))
&
Putting together all the previous Lemmas, write the main result of the work.
Theorem 3.9 (QEM for GMM). We assume have quantum access to GMM with parameters 7/' For parameters da Sps5r Er >> O, the running time of one iteration of the Quantum snnectaaioonMaxmmzaatli (QEM) algorithm is
O(Ta +Ta+ +T))).)
(D)()))) for To = Ö (13.5 (1..5,))
Tp = δ M (3311)))
Ts
63
ō(
((ad (V ) 2 "x (E))(E)
and T, = Ô 11 1.5 55 (5* (D))(2)
For the range of parameters of interest, the running time dominated by Tr.
The proof follows directly from the previous lemmas. Note that the cost of the whole algorithm is given by repeating the Estimation and the Maximization steps several times, until the threshold on the log- likelihoooo is reached. Note also that the expression of the runtime can be simplified from the observation that the cost of performing tomography on the covariance matrices Ej dominates the cost.
4 EXPERIMENTAL RESULTS
In this section, we present the results of some experiments on real datasets to bound the condition number and the other parameters of the runtime. Let's discuss the value of <(E), x(V), ((E), and v(V). We can thresholding the condition number by discarding small singular values of the matrix, as used in quantum linear algebra, might be advantagcous. This is indeed done often in classical machine learning models, since discarding the eigenvalues smaller than a certain threshold might even improve upon the metric under consideration (i.e. often the accuracy), and a form of regularization (Murphy 2012 Section 6.5). This is equivalent to limiting the eccentricity of the Gaussians. We can have similar consideration on the condition number of the dataset s<V). As shown before, the condition number of the matrix V' appearing in Lemma 3.2 is x²2 (V). Similarly,
claim that the value of (uV) will not increase significantly as we add vectors to the training set. Remember that we have some choice in picking the function μ:: in previous experiments we have found that choosing the maximum norm of the rows of lead to values of μ around 10, and also in this case we expect the samples of well-clusterable (Kerenidis et al. 2018) dataset to be constant. Also, j is bounded by the Frobenius norm of V. In case the matrix V can be clustered with high-enough accuracy by
7

---

k-means, it has been showed that the Frobenius norm of the matrix is proportional to vk. Given that EM is more powerful extension of k-means, we can rely on similar observations too. Usually, the number of features d is much more than the number of components in the mixture, i.e. d >>> k. so we expect de to dominate the 1.3. term in the cost needed to estimate the mixing weights. This makes the runtime of single iteration proportional to:
( (V) 83
As we said, the quantum running time saves the factor that depends on the number of samples and introduces a number of other parameters. Using our experimental results we can see that when the number of samples is large enough one can expect the quantum running time to be faster than the classical one. Note as well that one can expect to save some more factors from the quantum running time with more careful analysis.
Experiments. In the algorithm, we need to set the parameters 6, and óg to be small enough such that the likelihood is perturbed less than 7 7/4. We have reasons to believe that on well-clusterable data, the value of these parameters will be large enough, such as not to impact dramatically the runtime. A quantum version of k-means algorithm has already been simulated on the MNIST dataset the performances of q-means, the d-resistant version of the classical k-means algorithm. The experiment concluded that, for datasets that expected to be clustered nicely by this kind of clustering algorithms, the value of the parameters 6,, 60 did not decrease by increasing the number of samples nor the number of features. We expect similar behaviour the EM case, namely that for large datasets the impact on the runtime of the errors (6μ. 6.) does not cancel out the exponential gain in the dependence on the number of samples. For instance, in all the experiments of q-means (Kerenidis et al. 2018) on the MNIST dataset the value of 8, (which in their case was called just d) has been between 0.2 and 0.5. The value of c is usually (for instance in scikit-learn (Pedregosa et al. 2011 chosen to be 10 3 The value of 7 has always been below 11.
real data under similar assumptions (Kerenidis et al. 2018). There, the authors analyzed
We also analyzed some other real-world dataset, which can be fitted well with the EM algorithm (Reynolds et al. 2000 APPML. Voxforge.org) to perform speaker recognition: the task of recognizing a speaker from a voice sample, having access to a training set of recorded voices of all the possible speakers. Details of the measurements are reported in the Supplementary Material section, here we report only the results in Table T After this, we also experimented the impact of errors on the mixing weights in the accuracy of ML estimate of GMM by perturbing the trained model, by adding some random noise. With a value of óg = 0.035, 6,2 = 0.5 we correctly classified 98.2% utterances.
Σ 2
logdet( Σ)
k² (E)
2(
n(V)
<(V)
Acc. (%)
MAP
avg
0.244
58.56
4.21
3.82
2.14
23.82
99.4
2.45
70.08
50
4.35
2.79
40.38
ML
avg
1.31
14.56
15.57
2.54
2.14
23.82
98.8
max
3.44
92,3
50
3.67
2.79
40.38
Table 1: We estimate some of the parameters of the VoxForge (Voxforge.org) dataset. The averages for the matrix V are taken over 34 samples, while for Σ is over 170 samples. The accuracy reported in the experiments is measured on 170 samples in the test set, after the threshold on the eigenvalues of Σ. Each model is the result of the best of 3 different initializations of the EM algorithm. The first and the second column are the maximum singular values of all the covariance matrices, and the absolute value of the log-determinant. The column K (E) consist in the thresholded condition number for the covariance matrices.
In conclusion, the experimental results suggest that the influence of the extra parameters in the quantum running time (condition thresholds, errors, etc.) is moderate. This allows to be optimistic that, when quantum computers with quantum access to data become a reality, our algorithm (and improved versions that reduce even more the complexity with respect these extra parameters) could be useful in analyzing large datasets.
5 ACKNOWLEDGEMENT
We would like to thank the authors of (Miyahara et al. 2019) for sharing an early version of their manuscript with us. Part of this research was supported by the projects: ANR QuDATA and QuantERA QuantAlgo.
8

---

REFERENCES
Esma Aimeur, Gilles Brassard, and Sébastien Gambs, Quantum speed-up for unsupervised learning. Machine Learn- ing, 9022)261-287, 2013.
APPMI..
Spoken
speaker
identification based on
gaussian mixture
models
Python
implementation. accessed 20/07/2019.
www.aooonmmmmmeemmeem Amplementation/
Juan Miguel Arrazola, Alain Delgado, Bhaskar Roy Bardhan, and Seth Lloyd. Quantum-inspired algorithms in prac- tice. arXiv preprint urXiv:/905, 10415, 2019.
David Arthur and Sergei Vassilvitskii. k-means++: The advantages of careful seeding. Proceedings of the eigh- leenth annual ACM-SIAM symposium on Discrete algorithuss, pp. 1027-1035, Society for Industrial and Applied Mathematics, 2007.
Mohd Ali Balafar, Abdul Rahman Ramli, lqbal Saripan, and Syamsiah Mashohor. Review of brain mri image segmentation methods. Artificial Intelligence Review, 33(3):261-274, 2010.
Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549(7671):195, 2017.
Christophe Biernacki, Gilles Celeux, and Gérard Govaert. Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate gaussian mixture models. Computational Statistics & Data Analysis, 41(--4);
561-575, 2003.
Jeff A Bilmes et al. A gentle tutorial of the EM algorithm and its application parameter estimation for gaussian mixture and hidden markov models. 1998.
Johannes Blömer and Kathrin Bujna. Simple methods for initializing the EM algorithm for gaussian mixture models. CoRR, 2013.
Christos Boutsidis, Petros Drineas, Prabhanjan Kambadur, Eugenia-Maria Kontopoulou, and Anastasios Zouzias. A randomized algorithm for approximating the log determinant of symmetric positive definite matrix. Linear Algebra and its Applications, 533:95- 117, 2017.
Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum Amplitude Amplification and Estimation. Contemporary Mathematics, 305, 2002.
Gilles Celeux and Gérard Govaert. A classification EM algorithm for clustering and two stochastic versions. Compu- tational statistics & Data analysis, 14(3):315 33-2 1992.
Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. arXiv preprint 1804:008000999 2018
Nai-Hui Chia, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired sublinear classical algorithms for solving low-rank linear systems, arXiv preprint wwXXiv:811.0444 2018
Kenneth W Church and William A Gale. Poisson mixtures. Natural Language Engineering, 1(2): 1995.
Sanjoy Dasgupta. Learning mixtures of gaussians. In 40th Annual Symposiun Foundations of Computer Science (Cat. No. 99CB37039), pp. 634-644. IEEE, 1999.
Arthur Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the royal statistical society. Series B (methodological), pp. 1-38, 1977.
AR Dexter and DW Tanner. Packing densities of mixtures of spheres with log-normal size distributions. Nature physical science, 238(80):31, 1972.
Xiaodan Fan, Yuan Yuan, Jun S Liu, et al. The EM algorithm and the rise of computational biology. Statistical Science, 2764)4476-4 2010.
9

---

Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near temn processors. arXiv preprint arXiv: 1802.0600,, 2018.
ME Ghitany, Ross A Maller, and Zhou. Exponential mixture models with long-term survivors and covariates. Journal of multivariate Analysis, 49(2):218-2411 1994.
András Gilyén, Seth Lloyd, and Ewin Tang. Quantum-inspire low-rank stochastic regression with logarithmic de- pendence on the dimension. arXiv preprint arXiv: 1tt::8111..49 1.04909, 2018a.
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. arXiv preprint arXiv:/806.0/888 2018b.
Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum Algorithm for Linear Systems of Equations. Physical Review Letters, 103(15); 10 2009. ISSN 0031-90079 doi: 10.1 h00ss//hysseeleett11....00000
URLpt
//link. aps. wwtp://////oooooo/ http://o.....o0////eeeeeeeeeee 103. 150502
Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning. volume of Springer
Series in Statisticy. Springer New York, New York, NY, 2009. ISBN 978-0-387-84857-0. doi: 10. 10.1007//94608. com/ wndiinidxiiie 10. 1007////4940
URL http:: http::/wwwwwww.........ee
N Jiang, Y-F Pu, W Chang, Li, Zhang, and L-M Duan. Experimental realization of 105-qubit random access quantum memory. npj Quantum Information, 5(1): 2019.
Michael Keams, Yishay Mansour, and Andrew Y Ng. An information-theoreiic analysis of hard and soft assignment methods for clustering. In Learning in graphical models, pp. 495-520. Springer, 1998.
lordanis Kerenidis and Alessandro Luongo. Quantum classification of the MNIST dataset via Slow Feature Analysis. arXiv preprint a 088377 2018.
Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. Proceedings of the 8th Innovations in Theoretical Computer Science Conference, 2017a.
lordanis Kerenidis and Anupam Prakash. Quantum gradient descent for linear systems and least squares. arXiv preprint arXiv: 04992, 2017b.
lordanis Kerenidis and Anupam Prakash. A quantum interior point method for LPs and SDPs 0rXiv:1808.09266. 2018.
lordanis Kerenidis, Jonas Landman, and Anupam Luongo, Alessandro Prakash. q-means: A quantum algorithm for unsupervised machine learning. Accepted at NeurIPS 2019 arXiv preprint arXiv: 18/2.03584, 2018.
Reginald L Lagendijk, Jan Biemond, and Dick E Boekee. Identification and restoration of noisy blurred images using the petpccatiionmmaimiztttiii algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(7): 1180-1191, 1990.
Zhaokai Li, Xiaomei Liu, Nanyang Xu, and Jiangfeng Du Experimental realization of a quantum support vector machine. Physical review letters, 114(14):14504, 2015.
Bruce Lindsay. Mixture models: theory, geometry and applications. In NSF-CBMS regional conference series probability and statistics, pp. i-163. JSTOR, 1995.
Chuanhai Liu and Donald B Rubin. ML estimation of the distribution using EM and its extensions, ECM and ECME. Statistica Sinica, pp. 19-39, 1995.
Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum algorithms for supervised and unsupervised machine learning. arXiv, 1307.041 1:11::111111 7 2013a. URL htt.ttpp//aaxxvv.o///aa//////// 0011
Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum algorithms for superviise and unsupervised machine learning, arXiv, 330700411:1-11 7 2013b.
Hideyuki Miyahara, Kazuyuki Aihara, and Wolfgang Lechner. Quantum expectation supectation-maxxiizzaiion algorithm. Per- sonal Communication, 2019.
10

---

Kevin P Murphy. Machine learning: probabilistic perspective. MIT press, 2012.
Andrew Ng. Cs229 lecture notes machine learning. Lecture notes CS229 Stanford, 2012.
Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information, 2002.
JS Otterbach, R Manenti, N Alidoust, A Bestwick, M Block, B Bloom, S Caldwell, N Didier, E Schuyler Fried, S Hong, et al. Unsupervised machine learning on hybrid quantum computer. arXiv preprint arXiv:1712.05771, 2017.
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-leam: Machine learning in Python. Journal of Machine Learning Research, 12:2825-2830, 2011.
Douglas A Reynolds, Thomas F Quatieri, and Robert Dunn. Speaker verification using adapted gaussian mixture models. Digital signal processing, 10(1-3): 2000.
Walter Rudin et al. Principles of mathematical analysis, volume 3. McGraw-hill New York, 1964.
Yigit Subaşı, Rolando D Somma, and Davide Orsucci. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Physical review letters, 122(6):060504, 2019.
Ewin Tang. Quantum-inspired classical algorithms for principal component analysis and supervised clustering. arXiv preprint arXiv:1811.00417 2018a.
Ewin Tang. Quantum-inspiree classical algorithms for principal component analysis and supervised clustering, arXiv preprint 0rXiv:/811.00414 2018b.
Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. arXiv preprint arXiv: 1807 2018c.
Voxforge.org. Free speech... recognition voxforge.org
accessed 20/07/2019.
Nathan Wiebe, Ashish Kapoor, and Krysta M Svore. Quantum Algorithms for Neirest Methods for Super- vised and Unsupervised Learning. 2014. URL http:::/aax....og//oooeooo
2142. pd
Nathan Wiebe, Ram Shankar, and Siva Kumar. Hardening Quantum Machine Learning Against Adversaries. 2017.
Jianhua Yin and Jianyong Wang. A dirichlet multinomial mixture model-based approach for short text clustering. Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 233-242. ACM, 2014.
6 SUPPLEMENTARY MATERIAL
6.1 PREVIOUS RESULTS USED IN THE PROOFS
Here we report useful Theorems, Lemmas, and Claims that we use in the main text.
Theorem 6.1 (Determinant estimation (Boutsidis et al. 2017)). Let M Raxd be a positive definite matrix with eigenvalues in the interval (amin: 1). Then for all E 1) and >> O there is classical algorithm that outputs
log(del(M)) such that [l)) log(del( (M)) _ 2e| (ogle((ee((M)) with probability least 1 8 in time
TDet, == log(1/c)log() mnz(M))
c20min
Theorem 6.2 (Multivariate Mean Value Theorem (Rudin et al. 1964 Let U be an open set of Rdd, For differentiable functions f U + IR it holds that Vx, E U, 3c such that f(x) f() = Vf(c) (x - 1). Theorem 6.3 (Componentwise Softmax function (v(v) is Lipsehitz continuous). For d > 2, let o; Rd + (0, 1) be Then og is Lipschitz continuous, with K < V2.
the softmax, function defined as (y(v) Eisee
e'i
11

---

Algorithm 2 Expectation-Maxxiizaaioo for GMM
Require:
Dataset V. tolerance > 0.
Ensure:
GMM 7d (0', f ) that maximizes locally the likelihood f(y: V) up to tolerance т.
1: Select 70 (0", vt, ) using classical initialization strategies described in Subsection 6.3
t
repeat Expectation
j, calculate the responsibilities as:
0 03 11 Eng) 15 Σ Di 0f p4, x.)
(3)
5: Maximization
Update the parameters of the model as:
05-1 Σ
(4)
1=1
11 11t1 Σ r vi
(5)
Σ rig
Ejtl t E Ti (or Hj (1 )), - 10-1,,11 1
(6)
E rid
1=++1 until
- <
9: Return y²= -000 11, )
Proof. We need to find the such that for all z, Rd, we have that (,, (z))
Observing that o is differentiable and that if we apply Cauchy- Suuchy-Scwwrzz. to the statement of the Mean-Value- Theorem we derive that Vx, y U, 3e such that [=) - f(y)ll s [[)]( ||x yll- So show Lipschitz continuity it is enough
k z||.
select $ ||Vo,
max.eR" [V)le)]
The partial derivatives dayle)
du are o,(v)( (v)) if and -o,()),(()) otherwise. So || (-ov),,, (1-
< E -)))))))))))
(())a,(()) + (o((v(1- S s0 nd-1 oi(r) +1-
(u) $ (v) 2. In our we deduce that: (1,(()) (1)) & V2 =ll so K S V2.
Claim 6.4. (Kerenidis & Prakash 2017b) Let be the angle between vectors x, y, and assume that 9 x/2. Then. <2c |
||æ yll S c implies |||=:) - -1
We will also use Claim 4.5 from (Kerenidis et al. 2018).
Claim 6.5. (Kerenidis et al. 2018 Let e be the error we commit estimating (c) such that |||c) - -2)] < cb and ca the error we commit in the estimating the norms.
|||||| Tell S ea |c]]. Then ج|- cll S vT(e. + cb).
Definition 2 (Exponential Family
20122). probability density function or probubility mass function p p(olv) (u)
for v (V1. Um) ym, where V C IR, TRP is said to be in the exponential family if can be written as:
p(ul)) = h(v) exp(olu)TTeL - A(v)}
where:
RP is called the canonical natural parameter of the family,
12

---

o(v) is function of v (which often is just the identity function),
T(u) is the vector of sufficient statistics: a function that holds all the information the data holds with respect to the unknown parameters,
A(v) is the cumulant generating function, or log-partition finction,,, which acts as a normalization factor,
hu))> >> is the base measure which a non-informative prior and de-facto is scaling constant.
6.2 QUANTUM PROCEDURES
To prove our results, we are going to use the quantum procedures listed hereafter.
Theorem 6.6 (Amplitude estimation and amplification (Brassard et al. 2002)). If there is unitary operator U such that U 10) ( sin(0) |z. 0) + cos(0) |G, 0-) then (0n can be estimated to multiplicative error 1j in time ( T(U) and (z)) can be generated expected time O T(U)) ()in(())
sin(()
We also need some state preparation procedures. These subroutines needed for encoding vectors in e E IRd into quantum states Ju;). An efficient state preparation procedure is provided by the QRAM data structures. We stress the fact that our result continues to hold, no matter how the efficient quantum loading of the data is provided. For instance, the data can be accessed through a QRAM, through a block encoding, or when the data can be produced by quantum circuits.
Theorem 6.7 (QRAM data structure (Kerenidis & Prakash 2017at). Let V E Raxd, there is a data structure to store the rows of V such that,
The time insert, update or delete a single entry Viy O(log((n))
quantum algorithu with access to the data structure can perform the following unitaries time Olog" N).
(a) i) 10) + (1) [u.) for [n].
(b) 0) + Eie(n) ||ua|] (i).
In algorithm will also subroutines for quantum linear algebra. For symmetric matrix M gaxd with spectral norm ||M = stored in the QRAM, the running time of these algorithms depends linearly on the condition number x(M) of the matrix, that can be replaced by (,M,), a condition threshold where keep only the singular values bigger than t, and the parameter (MM), a matrix dependent parameter defined as
m(M) = min( (ain((|M|] (M)())) (MMM) PEP
for (p) == maj [m=l] where ||millp is the Ep norm of the i-th row of M, and P is a finte set of size 0(1) e
(0, 1]. Note that n(M) < || & /d as we have assumed that || = 1. The running time also depends
logarithmically on the relative error of the final outcome state. (Chakraborty et al. 2018 Gilyén al. 2018b).
Theorem 6.8 (Quantum linear algebra (Chakraborty et al. 2018, Gilyén et al. 2018b) ) Let M E Rdxd such that
|M and x E Rd Let e > If M is stored in appropriate QRAM data structures and the time to prepare (r) is Ta. then there exist quantum algorithms that with probability at least 1 - 1 polyy(d return state (z) such that
|||>) |Mx)|| S in time (((sMM))(MM) + (2)(M)) log(1/c)).
Theorem 6.9 (Quantum linear algebra for matrix products (Chakraborty et al. 2018) ). Let M2 E ddxdd such and x E IRd, and vector x E E. stored in QRAM. Let E > Then there exist quantum algorithms that with probability at least - 1/poly(d) returns a state (2) such that |||z) - MMx)|| I e in time (((M(M))((M))) + p(M2)TM..) log(1/e)), where TM,,Th TM,, is the time needed index the rows of M1 and M2.
that || ||
The linear algebra procedures above also be applied any rectangular matrix V E Nxd by considering instead the symmetric matrix V = VT
V) ))
13

---

The final component needed for the g-means algorithm is a linear time algorithm for vector state tomography that will be used to recover classical information from the quantum states corresponding to the new centroids in each step. Given a unitary U that produces a quantum state (x)), by calling Od d/c²) times U, the tomography algorithm is able to reconstruct a vector that approximates (x) such that [|]3) |=)|| S.
Theorem 6.10 (Vector state tomography (Kerenidis & Prakash 2018 Given access unitary U such that U (1) = la) and its controlled version in time TUU), there a tomography algorithm with time complexity (U) d alog d that produces unit vector I E Rd such that || - xll2 S с with probability at least (1 - 1/poly(d)).
Lemma 6.11 (Distance Inner Products Estimation (Kerenidis et al. 2018: Wiebe al. 2014 Lloyd al. 201 3a)). Assume for a data matrix V E IN xd and centroid matrix C' E R that the following unitaries 1) [0] H (c) (4)
and (i) (0) + l5) lej) can be performed in time T and the norms of the vectors are known. For any A > and O. there exists a quantum algorithm that computes
b6) 1b) 10)
+
}i) (5) (d=(us; (j)) where (d (z) - dl (vi,,y)) & with probability at least 1 - 2A, or
(6) (b) 10)
-
16) (á) (vi,,))), where (13, (g) - (vi;,s)] E e with probability at least 1 22
in time ① [I.]Ille, I|T 1og(1/4))
6.3 INITIALIZATION STRATEGIES FOR EM
Unlike k-means clustering, choosing good set of initial parameters for a mixture of Gaussian is by no means trivial, and multivariate context is known that the solution is problem-dependent. There are plenty of proposed techniques, and here we describe few of them. Fortunately, these initialization strategies can be directly translated into quantum subroutines without impacting the overall running time of the quantum algorithm.
The simplest technique is called random EM, and consists in selecting initial points at random from the dataset as centroids, and sample the dataset to estimate the covariance matrix of the data. Then these estimates are used as the starting configuration of the model, and we may repeat the random sampling until we get satisfactory results.
A more standard technique borrows directly the initialization strategy of k-means++ proposed in (Arthur & Vassil- vitskii, 2007), and extends it to make initial guess for the covariance matrices and the mixing weights. The initial guess for the centroids is selected by sampling from a suitable, easy to calculate distribution. This heuristic works as following: Let co be a randomly selected point of the dataset, first centroid. The other k 1 centroids are selected by selecting a vector U; with probability proportional to d (us, (((s)), where [[...)) is the previously selected centroid that is the closest U4 in lg distance. These centroids are then used initial centroids for a round of k-means algorithm to obtain 14 ...4, -Mj. Then, the covariance matrices can be initialized as 2g == Tej Liec, (t; - My (1, - m4,)² where Cj is the set of samples in the training set that have been assigned to the cluster j in the previous round of k-means. The mixing weights are estimated as Cj/n. Eventually X. is regularized to be PSD matrix.
There are other possible choices for parameter initialization in EM, for instance, based on Hierarchical Agglomerative Clustering (HAC) and the CEM algorithm. In CEM we run one step of EMM, but with so-called classification step be- tween E and M. The classification step consists in a hard-clustering after computing the initial conditional probabilities (in the E step). The M step then calculates the initial guess of the parameters (Celeux & Govaeri 1992). In the small EM initialization method we run EM with a different choice of initial parameters using some of the previous strategies. The difference here is that we repeat the EM algorithm for a few numbers of iterations, and we keep iterating from the choice of parameters that returned the best partial results. For an overview and comparison of different initialization techniques, we refer to (Blömer & Bujna 2013 Biernacki et al. (003).
Quantum initialization strategies For the initialization of 7 in the quantum algorithm we can use the same ini- tialization strategies as in classical machine learning. For instance, we can use the classical random EM initialization strategy for QEM.
A quantum initialization strategy can also be given using the k-means++ initializion strategy from (Kerenidis et al. 2018). which returns ie initial guesses for the centroids cf .....ch consistent with the classical algorithm in time 222
where E(d u1), is the average squared distance between two points of the dataset, and is the tolerance in the distance estimation. From there, we can perform full round of q-means algorithm and get an
2
E (v..e))
14

---

estimate for pΩ... p2. With q-means and the new centroids store in the QRAM we can create the state
(20) =
Σ 11) (((e)))
(8)
Where (1;)) is the label of the closest centroid to the i-th point. By sampling E O(d) points from this state we get two things. First, from the frequency fy of the second register we can have guess of f. < |Cyl/n fj/S. Then, from the first register we can estimate 2g. f- Eie(( 1))(s; - 14)².. Sampling ( points and creating the state in Equation (8) takes time ((dkn) by Theorem 6.11 and the minimum finding procedure described in (erenidis et 2018).
Techniques illustrated in (Miyahara al. 2019 can also be used quantize the CEM algorithm which needs hard- clustering step. Among the different possible approaches, the random and the small EM greatly benefit from a faster algorithm, we can spend more time exploring the space of the parameters by starting from different initial seeds, and thus avoid local minima of the likelihood.
6.4 SPECIAL CASES OF GMM.
What we presented in the previous section is the most general model of GMM. For simple datasets, is common to assume some restrictions on the covariance matrices of the mixtures. The translation into a quantum version of the model should be straightforward. We distinguish between these cases:
Soft k-means. This algorithm is often presented as a generalization of k-means, but it can actually be seen special case of EM for GMM albeit with different assignment rule. In soft k-means, the assignment function is replaced by a softmax function with stiffness parameter B3. This 33 represents the covariance of the clusters. It is assumed be equal for all the clusters, and for all dimensions of the feature space. Gaussian Mixtures with constant covariance matrix (i.e. E; = 31 for 3 E R) can be interpreted as kind of soft or fuzzy version of k-means clustering. The probability of point in the feature space being assigned certain cluster is:
e B|| =
Tij
D1=1 E 81
where B > 0 is the stiffness parameter.
Spherical. In this model, each component has its own covariance matrix, but the variance is uniform all the directions, thus reducing the covariance matrix to a multiple of the identity matrix (i.e. Σ, of for oj E IR).
Diagonal. As the name suggests, in this special case the covariance matrix of the distributions a diagonal matrix, but different Gaussians might have different diagonal covariance matrices.
4. Tied. In this model, the Gaussians share the same covariance matrix, without having further restriction on the Gaussian.
Full. This is the most general case, where each of the components of the mixture have a different, SDP, covariance matrix.
6.5 PROOFS
Lemma 6.12 (Determinant evaluation). There is an algorithm that, given as input a matrix Σ and a parameter < 1, outputs an estimate log (E)(()) such that
- log(det(()) S e with probability 1 in time:
T TDete = (((-~xx()) (20((////)2 log(det(E)))
Proof. In order to apply Theorem 6.1 we need to be sure that all the eigenvalues lie in (amin 1). In order to satisfy this condition, we can scale the matrix by a constant factor c, such that Σ' E/c. this way, log det(E') log (ar/c) Therefore, log(det(2") log(de((2)) log(c"). This will allow us to recover the value of log det( by using
15

---

Theorem 6.1 We apply the Theorem with precision = 1/4 to get an estimate y such that _ S log(det(E)) S 3 Then, 2' to have an estimate with absolute error e, we apply Theorem 6.1 with precision e' .7. This gives an estimate for log(det(>2) with error 2e' log(det(X)) 1 in time:
δ (c (c=2x(()) (og()/))nnz( (og(det(E))))
Lemma 6.13 (Error in the responsibilities of the exponential family). Let vs I be a vector, and let {p(1s|1z)}}= be a set of k probability distributions in the exponential family, defined as p(ualu;) == (vi)explo ( (T) A,(y))}. Then, ie have estimates for each exponent with error E, then can compute each rig such that (5j S 22k for j E [k].
Proof. The proof follows from rewriting the responsibility of Equation
3 as:
ry hy(1;) expfo - A,(1z) + log 1og
(9)
h((t:) - (r(i) + log 0/}
In this form, it is clear that the responsibilitiess be seen softmax function, and we can use Theorem 6.3 to bound the error in computing this value.
Let T₃ R&& be the vector of the exponent, that is tij (x())) A5(2z) + log 8.. In analogous way we define Ti the vector where each component is the estimate with error e. The error in the responsibility defined
as ray Tsyl [oy(T,) 0, (T])), Because the function o, is Lipschitz continuous. as we proved in Theorem
6.3 with Lipschitz constant K < 22, we have that, (T;) (Tii) s v2 [T Till. The result follows Vic.
T. T:|l
Lemma 6.14 (Quantum Gaussian Evaluation). Suppose we have stored in the QRAM a matrix V E Ruxd, the centroid μ E &a and the covariance matrix Rd of multivariate Gaussian distribution o( u., , E). as well as an estimate for log(det(E)) Then for 6 > 0, there exists a quantum algorithm that with probability 1 performs the mapping.
Ua..1 (8) 0) - |2) bt) such that |8 -.. < t1, where 8; = - - 11) T - 11) + d log 2n +
.
log(det(E))) the exponent for the Gaussian probability density function.
The running time of the al algorithm is,
k< ([))(()))og((/)) To.. = O
(10)
E1
Proof. We use quantum linear algebra and inner product estimation to estimate the quadratic form (v; (0---""---- p.) to error ej. First, we decompose the quadratic form as of'' each term in the sum to error e1/4.
22 Σ
"u+ p+++"* - 3-11 and separately approximate "D-1-+p
We describe the procedure to estimate μ -- = || wx|| |||||| (p 'v)) the other estimates are obtained sim- ilarly. We use the quantum linear algebra subroutines in Theorem 6.8 to construct |5--es) up to error 63 < 61 in time ((x(2)))(2))))()
which gives us the mapping [t) (0) + (t) 1l) We then quantum inner product estimation (Thcorem 6.11) to estimate Ju,--1 -1,;) to additive error -|| -. The procedure estimates (μ, Σ --,,), within additive error e The pro- and requires time O (()))(()) log(1/>) log(/e))))(()/s)) |||||| |||- -0a/-=/))) Using similar estimation procedure for of -- "u and μ Ty-T μu, we obtain an estimate for } ((v; μ1) -E -0: 1 14) within error c1.
and estimate ||>=>>;|| up to error é1 in time (()()))()))g(LLes))))))
cedure succeeds with probability
Recall that (through Lemma 3.1 we also have an estimate of the log-determinant to error Thus we obtain an approximation for -1((; - p1 > 1 (v. n) + d log 2 log(det(2)) within error 2e,. We have the upper bound,
16

---

1x='o;1] = || Ievll I x(E) |ell as ||ح|| = 1 for Σ stored in a QRAM data structure. Further observing that
||l|| I / and ||v/l| I .7. the running time for this computation is O (x)))( log(1/7) log(1/r2)))
Lemma 6.15 (Calculating responsibilities). Suppose we have quantum access GMM with parameters 7
(0', p², ). There are quantum algorithms that can:
1. Perform the mapping [t) (.) (0) ~ (8) (1) (r5) such that [T) ragl S e with probability --7 in time:
TR... = =O(k2 x To,e))
n1
12 Σ ==0
For given j E [b], construct state (Rz) such that RRj)
1
2
rij li)
< c
where Z3
rgj with
2z, ==0
high probability in time:
TR Ö(k² TRI,1))
Proof. For the first statement, let's recall the definition of responsibility: Túj En (10(v.ip With the aid of
UG of Lemma 3.2 we can estimate log(o(uiluı, Σj)) for all j up to additive error é1, and then using the current estimate of 0', we can calculate the responsibilities create the state,
(i> [lo(@ E,}> Tij j=1
The estimate Fzy is computed by evaluating weighted softmax function with arguments log(o(U, [1y, Egy for j [k]. The estimates log(é(v; (ovv:((()) [ 2),) are then uncomputed. The runtime of the procedure is given by calling & times Lemma for Gaussian estimation (the arithmetic operations to calculate the responsibilities
absorbed).
Let us analyze the error in the estimation of Tij. The responsibility Tey is softmax function with arguments log(p(ra))),.)) Σ,)) that are computed upto error é using Lemma 3.2 As the softmax function has a Lipschitz constant
K S 22 by Lemma 6.13 we choose precision for Lemma 3.2 to c//v2k get the guarantee [ii -ryll S c1. Thus, the total cost of this step is TRI, k1.5TG.
Now let's see how to encode this information the amplitudes, as stated in the second claim of the Lemma. We estimate the responsibilities rij to some precision e and perform a controlled rotation on an ancillary qubit obtain,
i)
Foy +
V1 1 11)).
(11)
We then undo the circuit on the second register and perform amplitude amplification on the rightmost auxiliary qubit being (0) to get (2y)
2 Tij
i). The runtime for amplitude amplification on this task is O(TR,.
y
R
Let us analyze the precision c required to prepare Ry) such that
RRy
R y) S 61. As we have estimates |rij Vnc. Applying Claim 6.4 the error for By the Cauchy-Schwarl inequality we have
Tayl < for all i,j, the norm error [ R
Σ" rij
i=
Tig
2
the normalized vector RR,) can be bounded as R,)
R2
<
[R
2nc
that ||
We can use this to obtain a bound
/ ||
n 0(k), using the dataset assumptions in
V
Σ. Fi
section
If we choose such that Vine < 61, that is S ///h /k then our runtime becomes Tao.e == Ō(k2 x TR).)). [R,
Lemma 6.16 (Computing (1-1). We assume quantum access to GMM with parameters 76 and let dg > 0 be a
precision parameter. There exists an algorithm that estimates 0* ++1
E RR such that |!0 -r+1 S 89 in time
Tg = O 4.35 1.5 k2 (E)
17

---

Proof. An estimate of 6--1 can be recovered from the following operations. First, we use Lemma 3.3 (part 1) to compute the responsibilities to error 61, and then perform the following mapping, which consists of a controlled rotation on an auxiliary qubit:
i) 1))
(v T2j 10) + (1-
nk
The previous operation has a cost of T R and the probability of getting [0) is p(0)
un Σ Ej-1 rig =
+
Recall that 0z+1 1 - by definition.
Let
Zj
Ei' and define state I/Bg)
Ei=1 ray (1) (0). After amplitude amplification on [0) we have the state,
1
VR)
R
Féj
1 (5)
Z
rig" 1)) l5)
Zz
R₂)
(12)
The probability of obtaining outcome (j) if the second register is measured in the standard basis is p(f) 8j
An estimate for 05+11 with precision can be obtained by either sampling the last register, by performing amplitude estimation to estimate each of the values g++1 for j E [k]] Sampling requires O(e)?) samples by the Chernoff bounds, but does not incur any dependence on k. In this case, as the number cluster k is relatively small compared to 1/c, we chose to do amplitude estimation to estimate all 0,+1 for j [k] to error e/ in time,
To 0 0 VKTRA
(13)
Let's analyze the error in the estimation of 0f+1 For the error due responsibility estimation by Lemma 3.3 we have
0, 07 0 R E.
Iray" riyl S é for all [ks), implying that g gr+ Vke The total error in & to error c/vk.
norm due to Amplitude estimation is at most estimates each coordinate B, ++1
Using the triangle incquality, we have the total error is at most + Vkej. As we require that the final error be upper bounded 0 ++ gt+1 dg. choose parameters viei 50/2 E1
Ae and e 60/2. With these 20h
parameters, the overall running time of the quantum procedure is To O(k 1.5 Tm O (13.5'
0 " (SIN(E)
82
Lemma 6.17 (Computing 11 We assume we have quantum access to a GMM with parameters J For a pre- cision parameter 8n > O, there is quantum algorithm that calculates {ي. ++ 1itttyn such that for all [k] ||a +1 11, 1+1 S 5, in time
Tg Ō (hddnvVVV))) + k35 2 (E)
Proof. The new centroid u5 is estimated by first creating an approximation of the state (R up to error e in the lz-
norm using part of Lemma 3.3 We then use the quantum linear algebra algorithms Theorem 6.8 to multiply Ry by
18

---

VT, and compute a state " along with an estimate for the norm step of the algorithm consists in estimating the unit vector |1z ) with precision Coom using tomography. Considering that the tomography depends on d, which we expect to be bigger than the precision required by the norm estimation, we can assume that the runtime of the norm estimation absorbed. Thus, we obtain: δ
|
with error Enorm- The last
h: foo
(VV (u(V) + TRa..)) ))
Let's now analyze the total error in the estimation of the new centroids, which we want to be 6 For this purpose, we use Claim 6.5 and choose parameters such that 2/TT(Etomm + Exorm) = Ōµ.- Since the error e3 for quantum linear algebra appears as a logarithmic factor in the running time, we can choose Eg < Étom without affecting the runtime.
Let yr be the classical unit vector obtained after quantum tomography, and (p} be the state produced by the quantum linear algebra procedure starting with an approximation of [Kt] Using the triangle inequality we have IIIx) - <
[m) |m)
< Ctom 61 < 6//2//7 The errors for the norm estimation procedure can be bounded
|x]l| Enorm + 61 S 6,/2//1- We therefore choose parameters ftom = 61 = Goorm S 0p/4/T Since the amplitude estimation step we use for estimating the norms does not depends on d, which is expected dominate the other parameters, we omit the amplitude estimation step. Substituting for T 62. we have the more concise expression for the running time of:
similarly ||
||||
|||||| -Pell |||
ō (vd))())((())) + k. 5 x² (())(2))
(14)
83
Lemma 6.18 (Computing ). We assume have quantum access GMM with parameters 7/. We also have computed estimates 15j" (+1 of all centroids such that || - 1 S 5 for precision parameter 6, > 0. Then, there exists a quantum algorithm that oulputs estimates for the new covariance matrices (E) 1 | E) +++1 11 s / with high probability, in time,
دزر such that
Ts: T=:= = ( (V) + 1 x= x (222)((2))
83
Proof. is simple to check, that the update rule of the covariance matrix during the maximization step can be reduced to (Murphy 2012. Exercise 11.2):
Mytl + Em (o
11 1T
Tijuoo n0;
1+1 (1Gt))T = Σ3 - - ++ (nf-1)r
(15)
11 E1 Téj
First, note that we can use the estimates of the centroids to compute n (uf with error 6, ||ll S 6 in the update rule for the Σj. This follows from the fact that z = μ + where is a vector of norm 6,. Therefore I|np" - p. < 52 S < It follows that we can allow an error of vTōō also for the left term in the definition of Eg.t.. Let's discuss the procedure for estimating Σ/ in Eq. (15). Note that vec|Eş] = (V')TR,
so we use quantum matrix multiplication to estimate |vec[X]]) and [[vec[]]]]]] As the runtime for the norm estima- log(1/mmat)) tion 2o does not depend on d, we consider it smaller than the runtime for performing tomography. Thus, the runtime for this operation is:
Snorms
d logd d 0
K (VV)(((V") + TR log(I/emutt)).
Tom
Let's analyze the error of this procedure. We want a matrix Σ that is -loo,e to the correct one: ド Σ
vee [X) vec| vec|x5]]|_ < vr Again, the error due to matrix multiplication can be taken as small as necessary,
19

---

since is inside a logarithm. From Claim 6.5 we just need to fix the error of tomography and norm estimation such that (uust + Enorms)
v where we have used 1) as an upper bound on
|>> F.. For the unit vectors, we require
Ctom + S 6 where Mj) is the error due to tomography |
and Egy is the error due to Lemma 3.3 For this inequality be true, we choose Ctom c1 6// /VT
The same argument applies to estimating the norm | with relative enror |||
زاا - Σ3
|| - |ا < c S 6//22/ (where here e the error of the amplitude estimation step used Theorem 6.8
and é is the error in calling Lemma 3.3 Again, we choose e1 S 6 /T. Note that k(V)) S x² (V).. This can be derived from the fact that <(A B) x (A))a()), ((BB) S k² x{A))((), and
[ei Eel²
V' =
(V&
(en 2 en)²)
Since the tomography is more costly than the amplitude estimation step, we can disregard the runtime for the norm estimation step. As this operation is repeated k times for the k different covariance matrices, the total runtime of the whole algorithm is given by δ kd (V)((" " xx(())(())))
Let us also note that for each new computed covariance matrices, we use Lemma 3.1 to compute an estimate for their -eg-determinant and this time can be absorbed in the time Tz.
Lemma 6.19 (Quantum estimation of likelihood). We assume we have quantum access GMM with parameters 7², For e > 0, there exists quantum algorithm that estimates Elp(van")) with absolute error c, in time
T: = Ō , 1.5 x² x²())(())
&
Proof. We obtain the likelihood from the ability to compute the value of a Gaussian distribution and quantum arith- metic. Using the mapping of Lemma 3.2 with precision 61, we can compute ((u:lyy, Σ,) for all the Gaussians, that is (i) j=o li) (g))- Then, by knowing 0, and by using quantum arithmetic we can compute a register the mixture of Gaussian's: p(007) = Eje(y) Opp(esli:7). We now drop the notation for the model and write p(e:) instead of p(v,)). Doing the previous calculations quantumly leads to the creation of the state (i) p(u.)). We perform the mapping i) Ip(x:) + Ii) (vp(e.)] + /1 p(u4) (1) and estimate p((0)) ~ E[p();) with amplitude estimation on the ancilla qubit. To get a esttimatee of p(0) we need to decide the precision parameter we use for estimating p(uali: and the precision required by amplitude estimation. Let p(0) be the c1-error intro- duced by using Lemma 3.2 and p(0) the error introduced by amplitude estimation. Using triangle inequality we set |P(0) p(0) <
p(0)- p(0) + p(0) p(0)| Er.
To have Ip(0) p(0) cr, should set é1 such that (p(0) p(0) < <,/4 and the error in amplitude estimation and in the estimation of the probabilities to be e2 The runtime of this procedure is therefore:
o k: To... vvp(o)) / = =ō - 1.5 <(E)a(E)
2
6.6 EXPERIMENTS
We used subset of the voices that can be found on VoxForge [Voxforge.org The training set consist in 5 speech utterances for 34 speakers (i.e. clips of few seconds of voice speech). From this data, common in the speech
20

---

recognition community to extract the so-called Mel Frequency Cepstrum Coefficients (MFCCs) features (Reynolds et al. 2000). We selected d 40 features for each speaker. Then, each speaker is modeled with a mixture of 16 different Gaussians. The test set consists of other 5 unseen utterances of the same 34 speakers (i.e. the training set and the test set have the same size). The task to correctly label the unseen utterances with the name of the correct speaker. This is done by testing each of the GMM fitted during the training with the new voice sample, and selecting the model with the highest likelihood. Due to the differences in the speakers' audio data, the different dataset V/... Vas are made of a variable number of points which ranges from 2000 to 4000.
In the vanilla configuration without thresholding, 169 among 170 utterances (5 utterances for all 34 speakers) were correctly labeled by EM with ML estimate, while all the elements in the test set were correctly recognized using the Bayesian (MAP) framework. During the training, we measured all the values of k(E), <<V) (V)),)() (=)) For almost all GMM fitted (choosing a diagonal covariance matrix), there is at least Σ, (among the 16 used model single speaker) with bad condition number (i.e up to 2500 circa). As in (Kerenidis et al. 2018 Kerenidis & Luongo 2018) we took a threshold on the matrix by discarding singular values smaller than certain value m. Practically, we discarded any singular value smaller than 2 x 10-2 Thresholding the covariance matrices did not impact the accuracy (the generalization error) significantly. In the MAP estimate, only one element is not correctly classified, while this number goes up to two in the case of ML estimates. The results are in Table
To check the resistence to noise, we perturbed each of the GMM 7², once fitted. Then, we measured variations of the accuracy on the test set. For each model, the perturbation consists of three things. First we add to f uniform vector (with randomly selected negative or positive entries) of Ez norm of 8g = 0.035. Then we perturb each centroid μı; with vector of norm smaller than 6, 0.5. In this error vector the sign of each noise component is chosen randomly, and the magnitude is sampled uniformly in the interval (0, )-)- Then, we perturbed also the diagonal matrices Σ, with a 10. As we are using a diagonal GMM, this reduces to perturbing each singular value by some random noise upper bounded by = Vd, making sure that each of the singular value stays positive, as covariance matrices are SPD. Last, the matrices are thresholded. Since the representation of the model used by our software stores the Cholesky's decomposition of the inverse, we worked with that representation. Notably, thresholding the Σ; help to mitigate the error of noise and regularize the model. With these parameters, we were able to correctly label 167 utterances over 170. We leave for the future further experiments with bigger and different types of datasets or where the noise is also added during the training process. We used scikit-learn (Pedregosa et al., 201T) to run all the experiments.
vector of norm smaller than T. where
6.7 QUANTUM MAP ESTIMATE OF GMM
Maximum Likelihood is not the only way to estimate the parameters of a model, and in certain cases might not even be the best one. For instance, in high- -igh-dimensioal spaces, it is pretty common for ML estimates to overfit. Moreover, it often the case that we have prior information on the distribution of the parameters, and we would like our models to take this information into account. These issues are often addressed using a Bayesian approach, i.e. by using so-called Maximum A Posterioni estimate (MAP) of a model (Murphy 2012 Section 14.4.2.8). MAP estimates work by assuming the existence of prior distribution over the parameters y The posterior distribution we use objective function to maximize comes from the Bayes' rule applied on the likelihood, which gives the posterior as product of the likelihood and the prior, normalized by the evidence. More simply, we use the Bayes' rule the likelihood
function, as p(7; V)= = p(V:))p(7) This allows us to treat the model y as a random variable, and derive from the MI. x(V)
estimate a MAP estimate:
YMP = arg max
log p(7/0g)
(16)
Among the advantages of a MAP estimate over MI that it avoids overfitting by having a kind of regularization effect on the model (Murphy 2012, Section 6.5). Another feature consists in injecting into a maximum likelihood model some external information, perhaps from domain experts. This advantage comes at the cost of requiring "good" prior information on the problem, which might be non-trivial. In terms of labelling, a MAP estimates correspond hard
21

---

clustering, where the label of the point is decided according to the following rule:
V/ = arg max Tij arg max log p(u; = 7:7) + p(ys = -):7)
(17)
Deriving the previous expression straightforward using the Bayes' rule, and by noting that the softmax is rank- preserving, and we can discard the denominator of ray since it does not depend on y - and it is shared among all the other responsibilities of the points U, Thus, from Equation 16 we can conveniently derive Equation [7 I7as as a proxy well. The Expectation
for the label. Fitting a model with MAP estimate is commonly done via the EM algorithm step of EM remains unchanged, but the update rules of the Maximization step are slightly different. In this work we only discuss the GMM case. for the other cases the interested reader is encouraged to see the relevant literature. For GMM, the prior on the mixing weight is often modeled using the Dirichlet distribution, that is 0, Dir(a)). For the rest of parameters, we assume that the conjugate prior is of the form p(uj, Σ₁ NIW (uj. Ln Eylma, la: Vy. So), where (IW Σÿ) is the Normal-inverse, Wishart distribution. The probability density function of the NIW is the product between multivariate normal @ppmo, ) and inverse Wishart distribution W v0). NIW has as support vectors with mean μl and covariance matrices !Σ where E is random variable with inverse Wishart distribution over positive definite matrices. NIW is often the distribution of choice in these cases, as is the conjugate prior of multivariate normal distribution with unknown mean and covariance matrix. A shorthand notation, let's define ry = n0, = E Tij- As in (Murphy 2012). we also denote with z and Sj +1 the Maximum Likelihood estimate of the parameters (11++2)xx ) and (>) F1FT) " For MAP, the update rules are the following:
g
r3 + 03 -1 n E3 aj - k
(18)
1+1 + T5 [+1 + momo
(19)
rj +
So + Sj ++
cory cotr
(xz 1+1 -mo) 1+1 mo)²
Djt
(20)
Vo rk dd+2
Where the matrix So is defined as:
So == bdld Di ,,
(21)
where each value 8j is computed as 8j E En x² which is the pooled variance for each of the dimension For more information on the advantages, vantaeas, and common choice of parameters of MAP estimate, we refer the interested reader to (Murphy) 2012). Using the QEM algorithm to fit a MAP estimate is straightforward, since once the ML estimate of the parameter is recovered from the quantum procedures, the update rules can be computed classically.
Corollary 6.20 (QEM for MAP estimates of GMM). We assume we have quantum access to a GMM with parameters Y. For parameters da, 5,, > 0, the running time of one iteration of the Quantum Maximum A Posteriori (QMAP) algorithm algorithm is
OTT, +T, T3 Tt).
22

---

for
To δ k. (2i))((22)
68
kclns(V) + 1... n² (E)(EE))
Tμ δ
6
Add (o + h 2 (5)((2))
Tx: o
Tt Ō k.s.s, ( (2))((()
For the range of parameters of interest, the running time is dominated by TE.
23


Reviews:
[{"review_id": "r1xbMt9AFH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors present a quantum algorithm for expectation maximization on a quantum machine, which is poly-logarithmic in the size of the dataset (and polynomial in other parameters such as the dimension of the feature space, and number of mixture components, condition number of the data and covariance matrices, some precision/error parameters etc) per iteration. So, compared to the . regular EM algorithm, this yields exponential speedup in the number of data points, but is worse in other factors (such as a k^4.5 dependence on the number of mixture components). So, the quantum system could be superior to a conventional computer for some settings of parameters. They run some (simulation) experiments on a dataset (VoxForge) to report accuracies (though there is no comparison of the accuracy of the classical algorithm). Also, there does not seem to be any experimental results to confirm the theoretical analysis of the scaling characteristics (i.e., to show that the scaling is as predicted by the theory).\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SJlz8BM9KS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Quantum machine learning is a hot topic, recently. There are several \"schools of thought\": \n-- quantum kernel embeddings, which work per-observation, and allow for application on noisy intermediate-scale devices,\n-- work based on quantum optimization solvers, which try to claim one could apply quantum eigensolvers to various training problems, but which does not seem to be applicable on intermediate-scale devices, given the scale of the requisite input,\n--  novel quantum algorithms. \nIn principle, the novel quantum algorithms hold the promise of an exponential speed-up. \n\nThe authors propose a novel variant of expectation minimisation for parameter estimation of a gaussian mixture model with uniform mixing coefficients, targeting an ill-defined model of quantum computing of their own coinage.  Unfortunately, the paper is very sloppy. \n\nThe sloppiness starts with the model of quantum computing, which seems to assume anything the authors needed:\n-- Definition 1: one can perform |k|d arithmetic operations in time polylog(d)\n-- sentence above Lemma 3.4: post-selection (which makes it possible to solve at least all of PP, cf. https://arxiv.org/abs/quant-ph/0412187)\n-- sentence below Lemma 3.6: \"quantum linear algebra subroutines and tomography\" in no time at all (?), wherein there are information-theoretic limits \\Omega(exp(kd)) on the state tomography.\n\nIn terms of statistics and learning theory, the authors:\n-- do not consider the separation of the Gaussians as a parameter. It is well-known that with o(1) separation, exponentially many samples are required, and with sqrt(log(k)) separation, polynomially many samples suggest (http://ieee-focs.org/FOCS-2017-Papers/3464a085.pdf). Unless Definition 1 of the authors \"subsumes\" processing of exponentially large numbers of samples in polylog(d) time, the authors may need to add an assumption on the separation.\n-- the authors explain that their approach works only for the gaussian mixture model with uniform mixing coefficients only at the top of page 4, in a completely obscure notation of their own, and do not reference it as an assumption in the theorems later. \n\nThe sloppiness continues with the description of the Experiments in Section 4. Authors present a table of some results, but do not mention whether these have been obtained on quantum-computing hardware, a simulator thereof (what simulator? with noise?), or whether this is some fully-classical variant of the algorithm being tested. This clearly violates any \"reproducibility checklist\".\n\nWithin the \"minor comments\" category:\n-- the introduction of the EM algorithms for GMM is sloppy. It is well known that EM for GMM is super-sensitive to noise and balance of the mixing coefficients (https://ieeexplore.ieee.org/document/8635825) and can get stuck in arbitrarily bad local optima, which is not mentioned once. \n-- the references to other algorithms for GMM mention only Dasgupta 1999, rather than the subsequent 20 years of research, e.g., of Ankur Moitra at MIT (https://math.mit.edu/directory/profile.php?pid=1502). \n-- on the other hand, there are plentiful references to arxiv pre-prints of Kerenidis et al, at least some of which have been shown to be vacuous, e.g., https://arxiv.org/abs/1808.09266 with an infinite upper bound on the run-time and no relation to the present paper? \n-- there are some formulae missing or ending half-way through, e.g. Page 3 before \"alas\", Page 3 after \"GMM would be\". \n-- there are a number of language issues: \"we can thresholding\", \"some other real-world dataset\".\n-- the discussion starting with \"Let's have a first high-level comparison\" is completely wrong. Especially the condition number estimates of 5 seem to have no justification what so ever. \n-- authors say that \"polynomial dependence on the rank, the error, and the condition number, make these algorithms impractical on interesting datasets\" -- but it is not clear whether they mean that their algorithm is also impractical?\n\nOverall, while I like the idea of parameter estimation on a quantum computer, I could not recommend accepting the paper in its current form.  ", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "N/A", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "HJgrkphAcS", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors present and analyze a quantum computing algorithm for learning GMMs.\n\nI think this paper cannot be accepted because it violates formatting guidelines. Also, I think it is not appropriate for ICLR since it assumes knowledge of quantum computing that most people at this conference would not have, and I as a reviewer do not possess, and hence cannot evaluate this paper. For example, I do not know bra-ket notation.\n\nIf the ACs disagree, I am happy to revise my review for this paper and try to be more thorough.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory.", "review_assessment:_checking_correctness_of_experiments": "I did not assess the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers were unanimous that this submission is not ready for publication at ICLR in its current form. 

Concerns raised include a significant lack of clarity, and the paper not being self-contained.
Decision: Reject