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Title: Neural Networks for Principal Component Analysis: A New Loss Function Provably Yields Ordered Exact Eigenvectors 

Paper content:
In this paper, we propose a new loss function for performing principal component analysis (PCA) using linear autoencoders (LAEs).
Optimizing the standard L2 loss results in a decoder matrix that spans the principal subspace of the sample covariance of the data, but fails to identify the exact eigenvectors.
This downside originates from an invariance that cancels out in the global map.
Here, we prove that our loss function eliminates this issue, i.e. the decoder converges to the exact ordered unnormalized eigenvectors of the sample covariance matrix.
For this new loss, we establish that all local minima are global optima and also show that computing the new loss (and also its gradients) has the same order of complexity as the classical loss.
We report numerical results on both synthetic simulations, and a real-data PCA experiment on MNIST (i.e., a 60,000 x784 matrix), demonstrating our approach to be practically applicable and rectify previous LAEs' downsides.
Ranking among the most widely-used and valuable statistical tools, Principal Component Analysis (PCA) represents a given set of data within a new orthogonal coordinate system in which the data are uncorrelated and the variance of the data along each orthogonal axis is successively ordered from the highest to lowest.
The projection of data along each axis gives what are called principal components.
Theoretically, eigendecomposition of the covariance matrix provides exactly such a transformation.
For large data sets, however, classical decomposition techniques are infeasible and other numerical methods, such as least squares approximation schemes, are practically employed.
An especially notable instance is the problem of dimensionality reduction, where only the largest principal components-as the best representative of the data-are desired.
Linear autoencoders (LAEs) are one such scheme for dimensionality reduction that is applicable to large data sets.
An LAE with a single fully-connected and linear hidden layer, and Mean Squared Error (MSE) loss function can discover the linear subspace spanned by the principal components.
This subspace is the same as the one spanned by the weights of the decoder.
However, it failure to identify the exact principal directions.
This is due to the fact that, when the encoder is transformed by some matrix, transforming the decoder by the inverse of that matrix will yield no change in the loss.
In other words, the loss possesses a symmetry under the action of a group of invertible matrices, so that directions (and orderings/permutations thereto) will not be discriminated.
The early work of Bourlard & Kamp (1988) and Baldi & Hornik (1989) connected LAEs and PCA and demonstrated the lack of identifiability of principal components.
Several methods for neural networks compute the exact eigenvectors (Rubner & Tavan, 1989; Xu, 1993; Kung & Diamantaras, 1990; Oja et al., 1992) , but they depend on either particular network structures or special optimization methods.
It was recently observed (Plaut, 2018; Kunin et al., 2019 ) that regularization causes the left singular vectors of the decoder to become the exact eigenvectors, but recovering them still requires an extra decomposition step.
As Plaut (2018) point out, no existent method recovers the eigenvectors from an LAE in an optimization-independent way on a standard network -this work fills that void.
Moreover, analyzing the loss surface for various architectures of linear/non-linear neural networks is a highly active and prominent area of research (e.g. Baldi & Hornik (1989) ; Kunin et al. (2019) ; Pretorius et al. (2018) ; Frye et al. (2019) ).
Most of these works extend the results of Baldi & Hornik (1989) for shallow LAEs to more complex networks.
However, most retain the original MSE loss, and they prove the same critical point characterization for their specific architecture of interest.
Most notably Zhou & Liang (2018) extends the results of Baldi & Hornik (1989) to deep linear networks and shallow RELU networks.
In contrast in this work we are going after a loss with better loss surface properties.
We propose a new loss function for performing PCA using LAEs.
We show that with the proposed loss function, the decoder converges to the exact ordered unnormalized eigenvectors of the sample covariance matrix.
The idea is simple: for identifying p principal directions we build up a total loss function as a sum of p squared error losses, where the i th loss function identifies only the first i principal directions.
This approach breaks the symmetry since minimizing the first loss results in the first principal direction, which forces the second loss to find the first and the second.
This constraint is propagated through the rest of the losses, resulting in all p principal components being identified.
For the new loss we prove that all local minima are global minima.
Consequently, the proposed loss function has both theoretical and practical implications.
Theoretically, it provides better understanding of the loss surface.
Specifically, we show that any critical point of our loss L is a critical point of the original MSE loss but not vice versa, and conclude that L eliminates those undesirable global minima of the original loss (i.e., exactly those which suffer from the invariance).
Given that the set of critical points of L is a subset of critical points of MSE loss, many of the previous work on loss surfaces of more complex networks likely extend.
In light of the removal of undesirable global minima through L, examining more complex networks is certainly a very promising direction.
As for practical consequences, we show that the loss and its gradients can be compactly vectorized so that their computational complexity is no different from the MSE loss.
Therefore, the loss L can be used to perform PCA/SVD on large datasets using any method of optimization such as Stochastic Gradient Descent (SGD).
Chief among the compellingly reasons to perform PCA/SVD using this method is that, in recent years, there has been unprecedented gains in the performance of very large SGD optimizations, with autoencoders in particular successfully handling larger numbers of high-dimensional training data (e.g., images).
The loss function we offer is attractive in terms of parallelizability and distributability, and does not prescribe any single specific algorithm or implementation, so stands to continue to benefit from the arms race between SGD and its competitors.
More importantly, this single loss function (without an additional post hoc processing step) fits seamlessly into optimization pipelines (where SGD is but one instance).
The result is that the loss allows for PCA/SVD computation as single optimization layer, akin to an instance of a fully differentiable building block in a NN pipeline Amos & Kolter (2017) , potentially as part of a much larger network.
In this paper, we have introduced a loss function for performing principal component analysis and linear regression using linear autoencoders.
We have proved that the optimizing with the given loss results in the decoder matrix converges to the exact ordered unnormalized eigenvectors of the sample covariance matrix.
We have also demonstrated the claims on a synthetic data set of random samples drawn from a multivariate normal distribution and on MNIST data set.
There are several possible generalizations of this approach we are currently working on.
One is improving performance when the corresponding eigenvalues of two principal directions are very close and another is generalization of the loss for tensor decomposition.
Before we present the proof for the main theorems, the following two lemmas introduce some notations and basic relations that are required for the proofs.
Lemma 2.
The constant matrices T p ∈ R p×p and S p ∈ R p×p are defined as
Clearly, the diagonal matrix T p is positive definite.
Another matrix that will appear in the formula- The following properties of Hadamard product and matrices T p and S p are used throughout:

assistant: text: A new loss function for PCA with linear autoencoders that provably yields ordered exact eigenvectors