Explaining the Explainer: A First Theoretical Analysis of LIME

01/10/2020 ∙ by Damien Garreau, et al. ∙ Université Nice Sophia Antipolis Universität Tübingen 0

Machine learning is used more and more often for sensitive applications, sometimes replacing humans in critical decision-making processes. As such, interpretability of these algorithms is a pressing need. One popular algorithm to provide interpretability is LIME (Local Interpretable Model-Agnostic Explanation). In this paper, we provide the first theoretical analysis of LIME. We derive closed-form expressions for the coefficients of the interpretable model when the function to explain is linear. The good news is that these coefficients are proportional to the gradient of the function to explain: LIME indeed discovers meaningful features. However, our analysis also reveals that poor choices of parameters can lead LIME to miss important features.

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1 Introduction

1.1 Interpretability

The recent advance of machine learning methods is partly due to the widespread use of very complicated models, for instance deep neural networks. As an example, the Inception Network

(Sze_Liu_Jia:2015) depends on approximately million parameters. While these models achieve and sometimes surpass human-level performance on certain tasks (image classification being one of the most famous), they are often perceived as black boxes, with little understanding of how they make individual predictions.

This lack of understanding is a problem for several reasons. First, it can be a source of catastrophic errors when these models are deployed in the wild. For instance, for any safety system recognizing cars in images, we want to be absolutely certain that the algorithm is using features related to cars, and not exploiting some artifacts of the images. Second, this opacity prevents these models from being socially accepted. It is important to get a basic understanding of the decision making process to accept it.

Model-agnostic explanation techniques aim to solve this interpretability problem by providing qualitative or quantitative help to understand how black-box algorithms make decisions. Since the global complexity of the black-box models is hard to understand, they often rely on a local point of view, and produce an interpretation for a specific instance. In this article, we focus on such an explanation technique: Local Interpretable Model-Agnostic Explanations (LIME, Rib_Sin_Gue:2016).

[scale=0.27]lime_for_images.pdf

Figure 1: LIME explanation for object identification in images. We used Inception (Sze_Liu_Jia:2015) as a black-box model. Terrapin, a sort of turtle, is the top label predicted for the image in panel (a). Panel (b) shows the results of LIME, explaining how this prediction was made. The highlighted parts of the image are the superpixels with the top coefficients in the surrogate linear model. We ran the same experiment for the ‘strawberry’ label in panel (c).

1.2 Contributions

Our main goal in this paper is to provide theoretical guarantees for LIME. On the way, we shed light on some interesting behavior of the algorithm in a simple setting. Our analysis is based on the Euclidean version of LIME, called “tabular LIME.” Our main results are the following:

  1. [noitemsep,topsep=0pt]

  2. When the model to explain is linear, we compute in closed-form the average coefficients of the surrogate linear model obtained by TabularLIME.

  3. In particular, these coefficients are proportional to the partial derivatives of the black-box model at the instance to explain. This implies that TabularLIME indeed highlights important features.

  4. On the negative side, using the closed-form expressions we show that it is possible to make some important features disappear in the interpretation, just by changing a parameter of the method.

  5. We also compute the local error of the surrogate model, and show that it is bounded away from  in general.

We explain how TabularLIME works in more details in Section 2. In Section 3, we state our main results. They are discussed in Section 4, and we provide an outline of the proof of our main result in Section 5. We conclude in Section 6.

2 LIME: Outline and notation

2.1 Intuition

From now on, we will consider a particular model encoded as a function and a particular instance to explain. We make no assumptions on this function, e.g., how it might have been learned. We simply consider  as a black-box model giving us predictions for all points of the input space. Our goal will be to explain the decision that this model makes for one particular instance .

As soon as  is too complicated, it is hopeless to try and fit an interpretable model globally, since the interpretable model will be too simple to capture all the complexity of . Thus a reasonable course of action is to consider a local point of view, and to explain  in the neighborhood of some fixed instance . This is the main idea behind LIME: To explain a decision for some fixed input , sample other examples around , use these samples to build a simple interpretable model in the neighborhood of , and use this surrogate model to explain the decision for .

One additional idea that makes a huge difference with other existing methods is to use discretized features of smaller dimension to build the local model. These new categorical features are easier to interpret, since they are categorical. In the case of images, they are built by using a split of the image  into superpixels (Ren_Mal:2003). See Figure 1 for an example of LIME output in the case of image classification. In this situation, the surrogate model highlights the superpixels of the image that are the most “active” in predicting a given label.

Whereas LIME is most famous for its results on images, it is easier to understand how it operates and to analyze theoretically on tabular data

. In the case of tabular data, LIME works essentially in the same way, with a main difference: tabular LIME requires a train set, and each feature is discretized according to the empirical quantiles of this training set.

[scale=0.25]general_setting.pdf

Figure 2: General setting of TabularLIME along coordinate . Given a specific datapoint (in red), we want to build a local model for  (in blue), given new samples (in black). Discretizing with respect to the quantiles of the distribution (in green), these new samples are transformed into categorical features  (in purple). In the construction of the surrogate model, they are weighted with respect to their proximity with  (here exponential weights given by Eq. (2.1), in black). In red, we plotted the tangent line, the best linear approximation one could hope for.

We now describe the general operation of LIME on Euclidean data, which we call TabularLIME. We provide synthetic description of TabularLIME in Algorithm 1, and we refer to Figure 2 for a depiction of our setting along a given coordinate. Suppose that we want to explain the prediction of the model  at the instance . TabularLIME has an intricate way to sample points in a local neighborhood of . First, TabularLIME constructs empirical quantiles of the train set on each dimension, for a given number of bins. These quantile boxes are then used to construct a discrete representation of the data: if falls between and , it receives the value . We now have a discrete version of , say . The next step is to sample discrete examples in uniformly at random: for instance, means that TabularLIME sampled an encoding such that the first coordinate falls into the first quantile box, the second coordinate into the third, etc. TabularLIME

subsequently un-discretizes these encodings by sampling from a normal distribution truncated to the corresponding quantile boxes, obtaining

new examples . For example, for sample we now sample the first coordinate from a normal distribution restricted to quantile box , the second coordinate from quantile box , etc. This sampling procedure ensures that we have samples in each part of the space. The next step is to convert these sampled points to binary features, indicating for each coordinate if the new example falls into the same quantile box as . Here, would be . Finally, an interpretable model (say linear) is learned using these binary features.

0:  Model , of new samples , instance , bandwidth , of bins , mean

, variance

1:   GetQuantiles(,,)
2:   Discretize(,)
3:  for  to  do
4:     for  to  do
5:         SampleUniform()
6:        
7:         SampleTruncGaussian()
8:        
9:     end for
10:     
11:  end for
12:  WeightedLeastSquares()
13:  return  
Algorithm 1 TabularLIME for regression

2.2 Implementation choices and notation

LIME is a quite general framework and leaves some freedom to the user regarding each brick of the algorithm. We now discuss each step of TabularLIME in more detail, presenting our implementation choices and introducing our notation on the way.

Discretization. As said previously, the first step of TabularLIME is to create a partition of the input space using a train set. Intuitively, TabularLIME produces interpretable features by discretizing each dimension. Formally, given a fixed number of bins , for each feature , the empirical quantiles are computed. Thus, along each dimension, there is a mapping associating each real number to the index of the quantile box it belongs to. For any point , the interpretable features are then defined as a vector corresponding to the discretization of  being the same as the discretization of . Namely, for all . Intuitively, these categorical features correspond to the absence or presence of interpretable components. The discretization process makes a huge difference with respect to other methods: we lose the obvious link with the gradient of the function, and it is much more complicated to see how the local properties of  influence the result of the LIME algorithm, even in a simple setting. In all our experiments, we took

(quartile discretization, the default setting).

[scale=0.30]quantiles_investigation.pdf

Figure 3: A visualization of the train set in dimension with , and . The empirical quantiles (dashed green lines) are already very close to the theoretical quantiles (green lines) for . The main difference in the procedure appears if  (red cross) is chosen at the edge of a quantile box, changing the way all the new samples are encoded. But for a train set containing enough observations and a generic , there is virtually no difference between using the theoretical quantiles and the empirical quantiles.

Sampling strategy. Along with , TabularLIME creates an un-discretization procedure . Simply put, given a coordinate  and a bin index , samples a truncated Gaussian on the corresponding bin, with parameters computed from the training set. The TabularLIME sampling strategy for a new example amounts to (i) sample

a random variable such that the

are independent samples of the discrete uniform distribution on

, and (ii) apply the un-discretization step, that is, return . We will denote by these new examples, and their discretized counterparts. Note that it is possible to take other bin boxes than those given by the empirical quantiles, the s are then sampled according to the frequency observed in the dataset. The sampling step of TabularLIME helps to explore the values of the function in the neighborhood of the instance to explain. Thus it is not so important to sample according to the distribution of the data, and a Gaussian sampling that mimics it is enough.

Assuming that we know the distribution of the train data, it is possible to use the theoretical quantiles instead of the empirical ones. For a large number of examples, they are arbitrary close (see, for instance, Lemma 21.2 in Van:2000). See Figure 3 for an illustration. It is this approach that we will take from now on: we denote the discretization step by  and denote the quantiles by for and to mark this slight difference. Also note that, for every , we set the quantiles bounding , that is, (see Figure 2).

Train set.

TabularLIME requires a train set, which is left free to the user. In spirit, one should sample according to the distribution of the train set used to fit the model . Nevertheless, this train set is rarely available, and from now on, we choose to consider draws from a

. The parameters of this Gaussian can be estimated from the training data that was used for 

if available. Thus, in our setting, along each dimension , the are the (rescaled) quantiles of the normal distribution. In particular, they are identical for all features. A fundamental consequence is that sampling the new examples s first and then discretizing has the same distribution as sampling first the bin indices s and then un-discretizing.

Weights.

We choose to give each example the weight

(2.1)

where is the Euclidean norm on and is a bandwidth parameter. It should be clear that  is a hard parameter to tune:

  • [noitemsep,topsep=0pt]

  • if  is very large, then all the examples receive positive weights: we are trying to build a simple model that captures the complexity of  at a global scale. This cannot work if  is too complicated.

  • if  is too small, then only examples in the immediate neighborhood of  receive positive weights. Given the discretization step, this amounts to choosing for all . Thus the linear model built on top would just be a constant fit, missing all the relevant information.

Note that other distances than the Euclidean distance can be used, for instance the cosine distance for text data. The default implementation of LIME uses instead of , with bandwidth set to . We choose to use the true Euclidean distance between  and the new examples as it can be seen as a smoothed version of the distance to  and has the same behavior.

Interpretable model.

The final step in TabularLIME is to build a local interpretable model. Given a class of simple, interpretable models , TabularLIME selects the best of these models by solving

(2.2)

where

is a local loss function evaluated on the new examples

, and is a regularizer function. For instance, a natural choice for the local loss function is the weighted squared loss

(2.3)

We saw in Section 1.1 different possibilities for . In this paper, we will focus exclusively on the linear models, in our opinion the easiest models to interpret. Namely, we set , with and . To get rid of the intercept , we now use the standard approach to introduce a phantom coordinate , and with and . We also stack the s together to obtain .

The regularization term is added to insure further interpretability of the model by reducing the number of non-zero coefficients in the linear model given by TabularLIME. Typically, one uses

regularization (ridge regression is the default setting of LIME) or

regularization (the Lasso). To simplify the analysis, we will set in the following. We believe that many of the results of Section 3 stay true in a regularized setting, especially the switch-off phenomenon that we are going to describe below: coefficients are even more likely to be set to zero when .

In other words, in our case TabularLIME performs

weighted linear regression

on the interpretable features s, and outputs a vector such that

(2.4)

Note that is a random quantity, with randomness coming from the sampling of the new examples . It is clear that from a theoretical point of view, a big hurdle for the theoretical analysis is the discretization process (going from the s to the s).

Regression vs. classification.

To conclude, let us note that TabularLIME can be used both for regression and classification. Here we focus on the regression mode: the outputs of the model are real numbers, and not discrete elements. In some sense, this is a more general setting than the classification case, since the classification mode operates as TabularLIME for regression, but with chosen as the function that gives the likelihood of belonging to a certain class according to the model.

2.3 Related work

Let us mention a few other model-agnostic methods that share some characteristics with LIME. We refer to Gui_Mon_Rug:2019 for a thorough review.

Shapley values.

Following Sha:1953 the idea is to estimate for each subset of features  the expected prediction difference when the value of these features are fixed to those of the example to explain. The contribution of the th feature is then set to an average of the contribution of  over all possible coalitions (subgroups of features not containing ). They are used in some recent interpretability work, see Lun_Lee:2017 for instance. It is extremely costly to compute, and does not provide much information as soon as the number of features is high. Shapley values share with LIME the idea of quantifying how much a feature contributes to the prediction for a given example.

Gradient methods.

Also related to LIME, gradient-based methods as in Bae_Sch_Har:2010 provide local explanations without knowledge of the model. Essentially, these methods compute the partial derivatives of  at a given example. For images, this can yield satisfying plots where, for instance, the contours of the object appear: a saliency map (Zei_Fer:2014). Shr_Gre_Shc:2016; Shr_Gre_Kun:2017 propose to use the “input derivative” product, showing advantages over gradient methods. But in any case, the output of these gradient based methods is not so interpretable since the number of features is so high. LIME gets around this problem by using a local dictionary with much smaller dimensionality than the input space.

3 Theoretical value of the coefficients of the surrogate model

We are now ready to state our main result. Let us denote by  the coefficients of the linear surrogate model obtained by TabularLIME. In a nutshell, when the underlying model  is linear, we can derive the average value  of the coefficients. In particular, we will see that the s are proportional to the partial derivatives . The exact form of the proportionality coefficients is given in the formal statement below, it essentially depends on the scaling parameters

and the s, the quantiles left and right of the s.

Theorem 3.1 (Coefficients of the surrogate model, theoretical values).

Assume that  is of the form , and set

(3.1)

where, for any , we defined

and

Let

. Then, with high probability greater than

, it holds that

A precise statement with the accurate dependencies in the dimension and the constants hidden in the result can be found in the Appendix (Theorem 10.1). Before discussing the consequences of Theorem 3.1 in the next section, remark that since is encoded by , the prediction of the local model at , , is just the sum of the s. According to Theorem 3.1, will be close to this value, with high probability. Thus we also have a statement about the error made by the surrogate model in .

Corollary 3.1 (Local error of the surrogate model).

Let . Then, under the assumptions of Theorem 3.1, with probability greater than , it holds that

with hidden constants depending on and the s.

Obviously the goal of TabularLIME is not to produce a very accurate model, but to provide interpretability. The error of the local model can be seen as a hint about how reliable the interpretation might be.

[scale=0.25]switch_off_1.pdf

Figure 4: Example where the true underlying black box model only depends on two features: . For each of the features, we plot the values of the s obtained by TabularLIME. The blue line shows the median over all experiments, the red cross the theoretical value according to our theorem. The boxplots contain values between first and third quartiles, the whiskers are times the interquartile ranges, and the black dots mark values outside this range. To produce the figure, we made repetitions of the experiment, with examples and . We see that TabularLIME finds nonzero coefficients exactly for the first two coordinates, up to noise coming from the sampling. This is the result that one would hope to achieve, and also the result predicted by our theory.

4 Consequences of our main results

We now discuss the consequences of Theorem 3.1 and Corollary 3.1.

Dependency in the partial derivatives.

A first consequence of Theorem 3.1 is that the coefficients of the linear model given by TabularLIME are approximately proportional to the partial derivatives of  at , with constant depending on our assumptions. An interesting follow-up is that, if  depends only on a few features, then the partial derivatives in the other coordinates are zero, and the coefficients given by TabularLIME for these coordinates will be  as well. For instance, if as in Figure 4, then , , and for all . In a simple setting, we thus showed that TabularLIME does not produce interpretations with additional erroneous feature dependencies. Indeed, when the number of samples is high, the coordinates which do not influence the prediction will have a coefficient close to the theoretical value  in the surrogate linear model. For a bandwidth not too large, this dependency in the partial derivatives seems to hold to some extent for more general functions. See for instance Figure 6, where we demonstrate this phenomenon for a kernel regressor.

[scale=0.25]tlime_tquantiles_lin_boston.pdf

Figure 5: Values of the coefficients obtained by TabularLIME on each coordinate in dimension for a linear model trained on the Boston housing dataset (Har_Rub:1978). The s are concentrated around the red crosses, which denote the s, the theoretical values predicted by Theorem 3.1. To produce the figure, we ran experiments with new samples generated for each run and we set .

Robustness of the explanations. Theorem 3.1 means that, for large , TabularLIME outputs coefficients that are very close to  with high probability, where  is a vector that can be computed explicitly as per Eq. (3.1). Still without looking too closely at the values of , this is already interesting and hints that there is some robustness in the interpretations provided by TabularLIME: given enough samples, the explanation will not jump from one feature to the other. This is a desirable property for any interpretable method, since the user does not want explanations to change randomly with different runs of the algorithm. We illustrate this phenomenon in Figure 5.

[scale=0.25]tlime_tquantiles_krr_boston.pdf

Figure 6: Values of the coefficients obtained by TabularLIME on each coordinate. We used the same settings as in Figure 5, but this time we train a kernel ridge regressor on the Boston Housing dataset—a nonlinear function. For the ridge regression, we used the Gaussian kernel with scale parameter set to  and default regularization constant (). We then estimated the partial derivatives of  at  and reported the corresponding s in red. For the chosen bandwidth (we took ), the experiments seem to roughly agree with our theory.
Influence of the bandwidth.

Unfortunately, Theorem 3.1 does not provide directly a founded way to pick 

, which would for instance minimize the variance for a given level of noise. The quest for a founded heuristic is still open. However, we gain some interesting insights on the role of 

. Namely, for fixed , , and , the multiplicative constants appearing in Eq. (3.1) depend essentially on .

Without looking too much into these constants, one can already see that they regulate the magnitude of the coefficients of the surrogate model in a non-trivial way. For instance, in the experiment depicted in Figure 4, the partial derivative of  along the two first coordinate has the same magnitude, whereas the interpretable coefficient is much larger for the first coordinate than the second. Thus we believe that the value of the coefficients in the obtained linear model should not be taken too much into account.

More disturbing, it is possible to artificially (or by accident) put to zero, therefore forgetting about feature  in the explanation, whereas it could play an important role in the prediction. To see why, we have to return to the definition of the s: since by construction, to have is possible only if

(4.1)

and is set to . We demonstrate this switching-off phenomenon in Figure 7. An interesting take is that  not only decides at which scale the explanation is made, but also the magnitude of the coefficients in the interpretable model, even for small changes of .

[scale=0.25]switch_off_2.pdf

Figure 7: Values of the coefficients given by LIME. In this experiment, we took exactly the same setting as in Figure 4, but this time set the bandwidth to instead of . In that case, the second feature is switched-off by TabularLIME. Note that it is not the case that  is too small and that we are in a degenerated case: TabularLIME still puts a nonzero coefficient on the first coordinate.
Error of the surrogate model.

A simple consequence of Corollary 3.1 is that, unless some cancellation happens between in the term , the local error of the surrogate model is bounded away from zero. For instance, as soon as , it is the general situation. Therefore, the surrogate model produced by TabularLIME is not accurate in general. We show some experimental results in Figure 8.

Finally, we discuss briefly the limitations of Theorem 3.1.

Linearity of .

The linearity of  is a quite restrictive assumption, but we think that it is useful to consider for two reasons.

First, the weighted nature of the procedure means that TabularLIME is not considering examples that are too far away from  with respect to the scaling parameter . Thus it is truly a local assumption on , that could be replaced by a boundedness assumption on the Hessian of  in the neighborhood of , at the price of more technicalities and assuming that  is not too large. See, in particular, Lemma 11.3 in the Appendix, after which we discuss an extension of the proof when  is linear with a second degree perturbative term. We show in Figure 6 how our theoretical predictions behave for a non-linear function (a kernel ridge regressor).

Second, our main concern is to know whether TabularLIME operates correctly in a simple setting, and not to provide bounds for the most general  possible. Indeed, if we can already show imperfect behavior for TabularLIME when  is linear as seen earlier, our guess is that such behavior will only worsen for more complicated .

[scale=0.25]tlime_tquantiles_local_perf.pdf

Figure 8: Histogram of the errors . The setting is the same as in Figure 4, but we repeated the experiment times. The red double arrow marks the value given by Corollary 3.1 around which the local error concentrate. With high probability, the error of the surrogate model is bounded away from .
Sampling strategy.

In our derivation, we use the theoretical quantiles of the Gaussian distribution along each axis, and not prescribed quantiles. We believe that the proof could eventually be adapted, but that the result would loose in clarity. Indeed, the computations for a truncated Gaussian distribution are far more convoluted than for a Gaussian distribution. For instance, in the proof of Lemma 

8.1 in the Appendix, some complicated quantities depending on the prescribed quantiles would appear when computing .

5 Proof of Theorem 3.1

In this section, we explain how Theorem 3.1 is obtained. All formal statements and proofs are in the Appendix.

Outline.

The main idea underlying the proof is to realize that is the solution of a weighted least squares problem. Denote by the diagonal matrix such that (the weight matrix), and set the response vector. Then, taking the gradient of Eq. (5.1), one obtains the key equation

(5.1)

Let us define and , as well as their population counterparts and . Intuitively, if we can show that and are close to and , assuming that is invertible, then we can show that is close to .

The main difficulties in the proof come from the non-linear nature of the new features , introducing tractable but challenging integrals. Fortunately, the Gaussian sampling of LIME allows us to overcome these challenges (at the price of heavy computations).

Covariance matrix.

The first part of our analysis is thus concerned with the study of the empirical covariance matrix . Perhaps surprisingly, it is possible to compute the population version of :

where the s were defined in Section 3, and  is a scaling constant that does not appear in the final result (see Lemma 8.1).

Since the s are always distinct from and , the special structure of  makes it possible to invert it in closed-form. We show in Lemma 8.2 that

We then achieve control of via standard concentration inequalities, since the new samples are Gaussian and the binary features are bounded (see Proposition 8.1).

Right-hand side of Eq. (5.1).

Again, despite the non-linear nature of the new features, it is possible to compute the expected version of in our setting. In this case, we show in Lemma 9.1 that

where the s were defined in Section 3. They play an analogous role to the s but, as noted before, they are signed quantities. As with the analysis of the covariance matrix, since the weights and the new features are bounded, it is possible to show a concentration result for  (see Lemma 9.3).

Concluding the proof.

We can now conclude, first upper bounding by

and then controlling each of these terms using the previous concentration results. The expression of  is simply obtained by multiplying and .

6 Conclusion and future directions

In this paper we provide the first theoretical analysis of LIME, with some good news (LIME discovers interesting features) and bad news (LIME might forget some important features and the surrogate model is not faithful). All our theoretical results are verified by simulations.

For future work, we would like to complement these results in various directions: Our main goal is to extend the current proof to any function by replacing  by its Taylor expansion at . On a more technical side, we would like to extend our proof to other distance functions (e.g., distances between the s and , which is the default setting of LIME), to non-isotropic sampling of the s (that is, not constant across the dimensions), and to ridge regression.

Acknowledgements

The authors would like to thank Christophe Biernacki for getting them interested in the topic, as well as Leena Chennuru Vankadara for her careful proofreading. This work has been supported by the German Research Foundation DFG (SFB 936/ Z3), and the Institutional Strategy of the University of Tübingen (DFG ZUK 63).

Bibliography

Supplementary material for:

Explaining the Explainer: A First Theoretical Analysis of LIME

In this supplementary material, we provide the proof of Theorem 3.1 of the main paper. It is a simplified version of Theorem 10.1. We first recall our setting in Section 7. Then, following Section 5 of the main paper, we study the covariance matrix in Section 8, and the right-hand side of the key equation (5.1) in Section 9. Finally, we state and prove Theorem 10.1 in Section 10. Some technical results (mainly Gaussian integrals computation) and external concentration results are collected in Section 11.

Bibliography

7 Setting

Let us recall briefly the main assumptions under which we prove Theorem 3.1. Recall that they are discussed in details in Section 2.2 of the main paper.

H 1 (Linear ).

The black-box model can be written , with and fixed.

H 2 (Gaussian sampling).

The random variables are i.i.d. .

Also recall that, for any , we set the weights to

(7.1)

We will need the following scaling constant:

(7.2)

which does not play any role in the final result. One can check that when , regardless of the dimension.

Finally, for any , recall that we defined

(7.3)

and

(7.4)

where are the quantile boundaries of . These coefficients are discussed in Section 5 of the main paper. Note that all the expected values are taken with respect to the randomness on the .

8 Covariance matrix

In this section, we state and prove the intermediate results used to control the covariance matrix . The goal of this section is to obtain the control of in probability. Intuitively, if this quantity is small enough, then we can inverse Eq. (5.1) and make very precise statements about .

We first show that it is possible to compute the expected covariance matrix in closed form. Without this result, a concentration result would still hold, but it would be much harder to gain precise insights on the s.

Lemma 8.1 (Expected covariance matrix).

Under Assumption 2, the expected value of is given by

Proof.

Elementary computations yield

Reading the coefficients of this matrix, we have essentially three computations to complete: , , and .

Computation of .

Since the s are Gaussian (Assumption 2) and using the definition of the weights (Eq. (7.1)), we can write

By independence across coordinates, the last display amounts to

We then apply Lemma 11.1 to each of the integrals within the product to obtain

We recognize the definition of the scaling constant (Eq. (7.2)): we have proved that .

Computation of .

Since the s are Gaussian (Assumption 2) and using the definition of the weights (Eq. (7.1)),

By independence across coordinates, the last display amounts to

Using Lemma 11.1, we obtain

We recognize the definition of the scaling constant (Eq. (7.2)) and of the coefficient (Eq. (7.3)): we have proved that .

Computation of .

Since the s are Gaussian (Assumption 2) and using the definition of the weights (Eq. (7.1)),

By independence across coordinates, the last display amounts to

Using Lemma 11.1, we obtain

We recognize the definition of the scaling constant (Eq. (7.2)) and of the alphas (Eq. (7.3)): we have proved that . ∎

As it turns out, we show that it is possible to invert in closed-form, therefore simplifying tremendously our quest for control of . Indeed, in most cases, even if concentration could be shown, one would not have a precise idea of the coefficients of .

Lemma 8.2 (Inverse of the covariance matrix).

If for any , then is invertible, and

Proof.

Define the vector of the s. Set , , , and

Then is a block matrix that can be written . We notice that

Note that, since  is an increasing function, the s are always distinct from  and . Thus

is an invertible matrix, and we can use the block matrix inversion formula to obtain the claimed result. ∎

As a direct consequence of the computation of

, we can control its largest eigenvalue.

Lemma 8.3 (Control of ).

We have the following bound on the operator norm of the inverse covariance matrix:

where .

Proof.

We control the operator norm of by its Frobenius norm: Namely,

where we used in the last step of the derivation. ∎

Remark 8.1.

Better bounds can without doubt be obtained. A step in this direction is to notice that is an arrowhead matrix (OLe_Ste:1996). Thus the eigenvalues of  are solutions of the secular equation

Further study of this equation could yield an improved statement for Lemma 8.3.

We now show that the empirical covariance matrix concentrates around . It is interesting to see that the non-linear nature of the new coordinates (the s) calls for complicated computations but allows us to use simple concentration tools since they are, in essence, Bernoulli random variables.

Lemma 8.4 (Concentration of the empirical covariance matrix).

Let and be defined as before. Then, for every ,

Proof.

Recall that : it suffices to show the result for the Frobenius norm. Next, we notice that the summands appearing in the entries of , , , and , are all bounded. Indeed, by the definition of the weights and the definition of the new features, they all take values in . Moreover, for given , they are independent random variables. Thus we can apply Hoeffding’s inequality (Theorem 11.1) to , , and . For any given , we obtain

We conclude by a union bound on the entries of the matrix. ∎

As a consequence of the two preceding lemmas, we can control the largest eigenvalue of .

Lemma 8.5 (Control of ).

For every , with probability greater than ,

Proof.

Let