. Several architectures and training protocols have proven to be very effective, including variational autoencoders (VAEs)
, generative adversarial networks (GANs) and normalising flows [53, 22].
In this paper we consider a generalization of some of these architectures to a continuous setting, where the samples to be generated belong to an infinite-dimensional function space. One reason is that many physical quantities of interest are better modeled as functions than vectors, e.g. solutions of partial differential equations (PDEs). In this respect, this work fits in the growing research area of neural networks in infinite-dimensional spaces, often motivated by the study of PDEs, which includes Neural Operators, Deep-O-Nets , PINNS  and many others.
A second reason concerns the promising applications of generative models in solving inverse problems. A typical inverse problem consists in the recovery of a quantity from noisy observations that are described by a ill-posed operator between function spaces 
. Virtually, every imaging modality can be modeled in such a way, including computed tomography (CT), magnetic resonance imaging (MRI) and ultrasonography. In recent years, machine learning based reconstruction algorithms have become the state of the art in most imaging applications[10, 46]. Among these algorithms, the ones combining generative models with classical iterative methods – such as the Landweber scheme – are very promising since they retain most of the explanaibility provided by inverse problems theory. However, despite the impressive numerical results [16, 57, 9, 34, 55, 11, 33], many theoretical questions have not been studied yet, for instance concerning stability properties of the reconstruction.
In this work (Section 2), we introduce a family of continuous generative neural networks (CGNNs), mapping a finite-dimensional space into an infinite-dimensional function space. Inspired by the architecture of deep convolutional GANs (DCGANs) , CGNNs are obtained by composing an affine map with several (continuous) convolutional layers with nonlinear activation functions. The convolutional layers are constructed as maps between the subspaces of a multi-resolution analysis (MRA) at different scales, and naturally generalize discrete convolutions. In our continuous setting, the scale parameter plays the role of the resolution of the signal/image. We note that wavelet analysis has played a major role in the design of deep learning architectures in the last decade [44, 17, 7, 19, 8].
The main result of this paper (Section 3) is a set of sufficient conditions that the parameters of a CGNN must satisfy in order to guarantee global injectivity of the network. This result is far from trivial because in each convolutional layer the number of channels is reduced, and this has to be compensated by the higher scale in the MRA. Generative models that are not injective are of no use in solving inverse problems or inference problems, or at least it is difficult to study their performance from the theoretical point of view. Furthermore, injective generators are needed whenever they are used as decoders in an autoencoder. In the discrete settings, some families of injective networks have been already thoroughly characterized [12, 41, 49, 24, 37, 50, 29, 28]. Note that normalizing flows are injective by construction, yet they are maps between spaces of the same (generally large) dimension, a feature that does not necessarily help with our desired applications.
Indeed, another useful property of CGNNs is dimensionality reduction. For ill-posed inverse problems, it is well known that imposing finite-dimensional priors improves the stability and the quality of the reconstruction [6, 14, 13, 5, 15], also working with finitely-many measurements [3, 31, 2, 4, 1]. In practice, these priors are unknown or cannot be analytically described: yet, they can be approximated by a (trained) CGNN. The second main result of this work (Section 4) is that an injective CGNN allows us to transform a possibly nonlinear ill-posed inverse problem into a Lipschitz stable one.
As a proof-of-concept (Section 5
), we show numerically the validity of CGNNs in performing image deblurring on the MNIST dataset with increasing levels of noise. The numerical model is obtained by training a VAE whose decoder is designed with a CGNN architecture. The classical Landweber iteration method is used as a baseline for comparison. We also provide some qualitative experiments on the expressivity of CGNNs.
2 Architecture of CGNNs
We first review the architecture of a DCGAN , and then present our continuous generalization. For simplicity, the analysis is done for D signals, but it can be extended to the D case (see Appendix A.4).
2.1 1D discrete generator architecture
A deep generative model can be defined as a map , where is a finite-dimensional space with , constructed as the forward pass of a neural network with parameters . Our main motivation being the use of generators in solving ill-posed inverse problems, we consider generators that carry out a dimensionality reduction, i.e. , that will yield better stability in the reconstructions.
As a starting point for our continuous architecture, we then consider the one introduced in . It is a map (we drop the dependence on the parameters
) obtained by composing an affine layer andconvolutional layers with nonlinear activation functions. More precisely:
which can be summarized as
The natural numbers are the vector sizes and represent the resolution of the signals at each layer, while are the number of channels at each layer. The output resolution is . Generally, one has , since the resolution increases at each level. Moreover, we impose that is divisible by for every . We now describe the components of .
Each layer includes a pointwise nonlinearity , i.e. a map defined as
The fully connected layer.
The first layer is , where is a linear map and is a bias term.
The convolutional layers.
The fractional-strided convolutional layerrepresents a convolution with stride such that , where is the number of input channels and the number of output channels, with . This convolution with stride corresponds to the transpose of the convolution with stride , and is often called deconvolution. We refer to Appendix A.2 for more details on fractional-strided convolutions.
Note that the most significant dimensional increase occurs in the first layer, the fully connected one. Indeed, after the first layer, in the fractional-strided convolutional layers the increase of the vectors’ size is compensated by the decrease of the number of channels; see Figure 1 for an illustration. At each layer, the resolution of a signal increases thanks to a deconvolution with higher-resolution filters, as we explain in detail in Appendix A.2. The final output is then a single high-resolution signal.
2.2 1D CGNN architecture
We now describe how to reformulate this discrete architecture in the continuous setting, namely, by considering signals in . The resolution of these continuous signals is modelled through wavelet analysis. Indeed, the higher the resolution of a signal, the finer the scale of the space to which the signal belongs. The idea to link multi-resolution analysis to neural networks is partially motivated by scattering networks .
In the discrete formulation, the intermediate spaces , with , describe vectors of increasing resolution. In the continuous setting, it is natural to replace these spaces by using a MRA of [20, 32, 43] (see Definition 1 in Appendix A.1), namely, using the spaces
with , representing an increasing (finite) sequence of scales. We have that if and only if , so that contains signals at a resolution that is twice that of the signals in . Thus, the relation between the indexes and is
where is a free parameter, or, equivalently,
Similarly to the discrete case, the intermediate spaces are for , with . The norm in these spaces is
The nonlinearities act on functions in by pointwise evaluation:
Note that this map is well defined if there exists such that for every . Indeed, in this case, for . Moreover, if a.e., then a.e. It is worth observing that, in general, this nonlinearity does not preserve the spaces , namely, . However, in the case when the MRA is associated to the Haar wavelet, the spaces consist of dyadic step functions, and so they are preserved by the action of .
The fully connected layer.
The map in the first layer is given by
where is a linear map and .
The convolutional layers.
We first need to model the stride in the continuous setting. A convolution with stride that maps functions from the scale to the scale with filter can be seen as the map
where denotes the continuous convolution and denotes the orthogonal projection onto the closed subspace . In other words,
As a consequence, the corresponding deconvolution (i.e. a convolution with stride ) is given by its adjoint, which can be easily computed since projections and convolutions are self-adjoint:
We are now able to model a convolutional layer. The -th layer of a CGNN, for , is
where is the nonlinearity defined above and are the convolutions with stride . In view of the above discussion, and of the discrete counterpart explained in Appendix A.2, we define
where the convolution is given by
with filters and biases .
Altogether, the full architecture in the continuous setting may be written as
which can be summarized as
A simple example: the Haar case.
Let be the scaling spaces of piecewise constant functions, i.e. consider an MRA with the Haar scaling function . This simple scaling function, in addition to naturally extending the discrete case to the continuous one, also makes it possible to simplify the structure of a CGNN. Indeed , thanks to the form of and the fact that and , defined as , have disjoint support for every . In this setting, the projections after the nonlinearities can be removed.
3 Injectivity of CGNNs
We are interested in studying the injectivity of the continuous generator (6) to guarantee uniqueness in the representation of the signals. The injectivity will also allow us, as a by-product, to obtain stability results for inverse problems using generative models, as in Section 4.
We consider here the 1D case with stride ; then, applying (2) iteratively, we obtain for . We also consider non-expansive convolutional layers, i.e. . We note that the same result holds also with expansive convolutional layers, arbitrary stride (possibly dependent on ) and in the 2D case (see Appendix A.4).
We make the following assumptions.
Assumptions on the scaling spaces
Remark 1 (Haar and Daubechies scaling functions).
For positive functions, such as the Haar scaling function, i.e. , condition (9) is easily satisfied. For the Daubechies scaling functions with
vanishing moments for( corresponds to the Haar scaling function), we verified condition (9) numerically. We believe that this condition is satisfied for every scaling function , but have not been able to prove this rigorously.
Assumptions on the convolutional filters
The following hypothesis asks that, at each convolutional layer, the filters are compactly supported with the same support, where represents the filters’ size. Generally, the convolutional filters act locally, so it is natural to assume that they have compact support. Furthermore, we ask the filters to be linearly independent, in a suitable sense: this is needed for the injectivity of the convolutional layers.
Let . For every , the convolutional filters of the -th convolutional layer (5) satisfy
where , and , where is the matrix defined by
The condition is sufficient for the injectivity of the convolutional layers, but not necessary. The necessary condition is given in Appendix A.3, and consists in requiring the rank of a certain block matrix to be maximum, in which is simply the first block. We note that this condition is independent of the scaling function , but depends only on the filters’ coefficients .
Remark 3 (Analogy between continuous and discrete case).
Assumptions on the nonlinearity
For simplicity, we consider the same nonlinearity in each layer (the generalization to the general case is straightforward). The following conditions guarantee that is injective.
We assume that
is injective and for every , for some ;
and preserves the sign, i.e. for every ;
Note that these conditions ensure that for every , for some , and so for every . It is also straightforward to check that the injectivity of ensures the injectivity of
In the Haar case, the projection after the nonlinearity can be removed, as explained at the end of Section 2.2, and we need to verify only the injectivity of instead of that of . As noted above, a sufficient condition to guarantee the injectivity of is the injectivity of . So, in the Haar case, Hypothesis 3 can be relaxed and replaced by:
There exists such that for every .
Hypothesis 3 is satisfied for example by the function
. Its relaxed version, in the Haar case, is satisfied by some commonly used nonlinearities, such the Sigmoid, the Hyperbolic tangent, the Softplus, the Exponential linear unit (ELU) and the Leaky rectified linear unit (Leaky ReLU). Our approach does not allow us to consider non-injective’s, such as the ReLU .
For simplicity, in Hypothesis 3, we require that and that is strictly positive everywhere. This allows us to use Hadamard’s global inverse function theorem  to obtain the injectivity of the generator. However, thanks to a generalized version of Hadamard’s theorem , we expect to be able to relax the conditions by requiring only that is Lipschitz and its generalized derivative is strictly positive everywhere. In this way, the Leaky ReLU would satisfy the assumptions.
Assumptions on the fully connected layer
We impose the following natural hypothesis on the fully connected layer.
We assume that
The linear function is injective;
There exists such that and .
The inclusion is natural, since we start with low-resolution signals. The second condition in Hypothesis 4 means that the image of the first layer, , contains only compactly supported functions with the same support. This is natural since we deal with signals of finite size.
Even the injectivity of is non-restrictive, since we choose the dimension of the latent space to be much smaller than the dimension of , which is . So, maps a low-dimensional space into a higher dimensional one.
The injectivity theorem
Sketch of the proof.
4 Stability of inverse problems with generative models
We now show how an injective CGNN can be used to solve ill-posed inverse problems. The purpose of a CGNN is to reduce the dimensionality of the unknown to be determined, and the injectivity is the main ingredient to obtain a rigurous stability estimate.
We consider an inverse problem of the form
where is a possibly nonlinear map between Banach spaces, and , modeling a measurement (forward) operator, is an quantity to be recovered and is the noisy data. Typical inverse problems are ill-posed (e.g. CT, accelerated MRI or electrical impedance tomography), meaning that the noise in the measurements is amplified in the reconstruction. For instance, in the linear case, this instability corresponds to having an unbounded (namely, not Lipschitz) inverse . The ill-posedness is classically tackled by using regularization, which often leads to an iterative method, as the gradient-type Landweber algorithm . This can be very expensive if has a large dimension.
However, in most of the inverse problems of interest, the unknown can be modeled as an element of a low-dimensional manifold in . We choose to use a generator to perform this dimensionality reduction and therefore our problem reduces to finding such that
In practice, the map is found via an unsupervised training procedure, starting from a training dataset. From the computational point of view, solving (13) with an iterative method is clearly more advantageous than solving (12), because belongs to a lower dimensional space. We note that the idea of solving inverse problems using deep generative models has been considered in [16, 55, 9, 34, 33, 46, 57, 11].
The dimensionality reduction given by the composition of the forward operator with a generator as in (13), has a regularizing/stabilizing effect that we aim to quantify. More precisely, we show that an injective CGNN yields a Lipschitz stability result for the inverse problem (13); in other words, the inverse map is Lipschitz continuous, and noise in the data is not amplified in the reconstruction. For simplicity, we consider the D case with stride and non-expansive convolutional layers, but the result can be extended to the D case and arbitrary stride as done in Appendix A.4 for Theorem 1.
5 Numerical results
We present here numerical results validating our theoretical findings. In Section 5.1 we describe how we train a CGNN, in Section 5.2 we apply a CGNN-based reconstruction algorithm to image deblurring, and in Section 5.3 we show qualitative results for generation and reconstruction purposes. Additional numerical simulations are included in Appendix A.6.
The conditions for injectivity given in Theorem 1
are not very restrictive and we can use an unsupervised training protocol to choose the parameters of a CGNN. Even though our theoretical results concern only the injectivity of CGNNs, we numerically verified that training a generator to also well approximate a probability distribution gives better reconstructions for inverse problems. For this reason, we choose to train CGNNs as parts of variational autoencoders (VAEs), a popular architecture for generative modeling. In particular, our VAEs are designed so that the corresponding decoder has a CGNN architecture. We refer to  for a thorough review of VAEs and to Appendix A.6 for more details on the numerical implementation. Note that there is growing numerical evidence showing that an untrained convolutional network is competitive with trained ones, when solving inverse problems with generative priors [57, 11].
The training is done on images in all our examples. We use the MNIST dataset of handwritten digits , which contains images in the training set and in the test set. Each example is a grayscale image. Since our training is unsupervised, we do not use any label information.
Our VAE includes a decoder with non-expansive convolutional layers and an encoder with a similar, yet mirrored, structure. More precisely, the decoder is composed by a fully connected layer, which maps vectors from the latent space (for different ) to channels of size . Then, there are three convolutional layers with stride . In each layer the size of the images doubles both in width and in height, while the number of channels is divided by . We eventually obtain one image of size , from which we extract a image, by removing the external rows and columns. At each layer, the same pointwise nonlinearity is applied.
The training is done with the Adam optimizer using a learning rate of
and the loss function, commonly used for VAEs, given by the weighted sum of two terms: the Binary Cross Entropy (BCE) between the original and the generated images and the Kullback-Leibler Divergence (KLD) between the standard Gaussian distribution and the one generated by the encoder in the latent space111All computations were implemented with Python3, running on a workstation with 256GB of RAM and 2.2 GHz AMD EPYC 7301 CPU and Quadro RTX 6000 GPU with 22GB of memory. All the codes are available at https://github.com/ContGenMod/Continuous-Generative-Neural-Network.
The injectivity of the decoder, i.e. the generator, is guaranteed if the hypotheses of Theorem 1 are satisfied, where Hypothesis 3 can be relaxed as in Remark 4, since, in practice, we are in the Haar case. We test Hypothesis 2 a posteriori, i.e. after the training, and, in our cases, it is always satisfied.
5.2 Deblurring with generative models
In Figure 2 we numerically show that combining Landweber iterations with a CGNN provides better and more stable results with respect to the standard Landweber scheme for a classical imaging inverse problem: image deblurring with a Gaussian blurring filter. In other words, we solve (13) instead of (12), where is a CGNN. More details on both algorithms can be found in Appendix A.6.
The ground truth is a handwritten digit taken from the MNIST test set. The data is obtained by blurring the ground truth with a fixed Gaussian blurring operator at four different levels of additive Gaussian noise. We compare the original image, the corrupted one , the reconstruction obtained by Landweber iterations , and the one obtained by our approach combining Landweber with a CGNN
, by measuring the peak signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM). We choose a CGNN with the architecture explained in Section5.1 with the Leaky ReLU activation function and the latent space dimension . For all noise levels, our CGNN based algorithm clearly outperforms the classical one with respect to both indicators, even though at low levels of noise the Landweber method recovers higher resolution details. We verified that choosing different activation functions does not change the reconstruction quality significantly.
To assess the quality of the generation, in Figures 2(a) and 2(b) we show random samples of images from trained D injective CGNN with different activation functions (see Appendix A.6). We consider the ReLU, which is not injective, the Leaky ReLU, which is injective but not , the ELU function, which is injective, but does not satisfy for some (see Condition 1 of Hypothesis 3), and a nonlinearity satisfying Hypothesis 3: . In order to improve the quality of the generation of new samples, we reweight the loss function as . We compare two different latent space dimensions: and . We qualitatively observe that has a generative power comparable to other common nonlinearities.
To evaluate the reconstruction power of our trained VAE, we compute the mean and the variance, over theimages of the test set, of the SSIM between the true image and the reconstructed one, obtained by applying the full VAE to the true image. Here, we fix the nonlinearity and the loss function and we consider different latent space dimensions (see Figure 2(c)). We observe that the SSIM increases significantly up to .
Left: random samples from trained injective CGNNs with different activation functions. Right: average (with standard deviation error bars) SSIM overimages from the test set, for several choices of the latent spaces dimension .
In this work, we have introduced CGNNs, a family of generative models in the continuous, infinite-dimensional, setting, generalizing popular architectures such as DCGANs . We have shown that, under natural conditions on the weights of the networks and on the nonlinearity, a CGNN is globally injective. This allowed us to obtain a Lipschitz stability result for (possibly nonlinear) ill-posed inverse problems, with unknowns belonging to the manifold generated by a CGNN.
The main mathematical tool used is wavelet analysis and, in particular, a multi-resolution analysis of . While wavelets yield the simplest multi-scale analysis, they are suboptimal when dealing with images. So, it would be interesting to consider CGNNs with other systems, such as curvelets  or shearlets , more suited to higher-dimensional signals.
For simplicity, we considered only the case of a smooth nonlinearity : we leave the investigation of Lipschitz ’s to future work. This would allow for including many commonly used activation functions. Some simple illustrative numerical examples (only in the case of the Haar wavelet) are included in this work, which was mainly focused on the theoretical properties of CGNNs. It would be interesting to perform more extensive numerical simulations in order to better evaluate the performance of CGNNs, also with other types of wavelets, e.g. with the Daubechies wavelets, and with nonlinear inverse problems, such as electrical impedance tomography.
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA8655-20-1-7027. The authors are members of the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni”, of the “Istituto Nazionale di Alta Matematica”.
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Appendix A Appendix
a.1.1 1D wavelet analysis
We give a brief review of concepts from wavelet analysis: in particular the definitions and the meaning of scaling function spaces and Multi-Resolution analysis in the D case. See [20, 32, 43] for more details.
Given a function , we define
for every . The integers and are the scale and the translation parameters, respectively, where the scale is proportional to the speed of the oscillations of (the larger , the finer the scale, the faster the oscillations).
A Multi-Resolution Analysis (MRA) is an increasing sequence of subspaces defined for
together with a function such that
is dense in and ;
if and only if ;
and is an orthonormal basis of .
The function is called scaling function of the MRA.
From Points and of Definition 1, we have that is an orthonormal basis of .
Given a function with the properties below, it is possible to construct an MRA with as scaling function. Indeed, if we require that
is a convergent sum in and ;
is continuous in with ;
then together with define an MRA.
Intuitively, the space contains functions where the finest scale is , which explains the name MRA.
a.1.2 2D Wavelet analysis
From [43, Section ], we recall the principal concepts of D wavelet analysis. In D, the scaling function spaces become with , with orthonormal basis given by , where
and in an orthonormal basis of . We recall that , where . As in D, the MRA properties of Definition 1 hold:
is dense in and ;
if and only if ;
and is an orthonormal basis of .
Moreover, for every , as in D.
a.2 Discrete strided convolutions
A fractional-strided convolution with stride such that , input channels and output channels is defined by
where are the convolutional filters and are the bias terms, for and . The operator is defined as
where we extend the signals and to finitely supported sequences by defining them zero outside their supports, i.e. , where is the space of sequences with finitely many nonzero elements. We can rewrite (16) in the following way:
where with , and for every . The symbol represents the discrete convolution
The output belongs to . We motivate (16) by taking the adjoint of the strided convolution with stride , as we explain below.
Equation (17) is useful to interpret Hypothesis 2 on the convolutional filters (Section 3). We observe that in our case the convolution is well defined since the signals we consider have a finite number of non-zero entries. However, in general, it is enough to require that and with to obtain a well-defined discrete convolution.
We now want to justify (16). We compute the adjoint of the convolutional operator with stride and filter , which is defined as , where
As before, the signals and are seen as elements of by extending them to zero outside their supports and the symbol denotes the discrete convolution defined in (18). The adjoint of , , satisfies
where is the scalar product on . Using (20), we find that
Figure 4 presents a graphical illustration of three examples of strided convolutions with different strides: in Figure 3(a), in Figure 3(b), and in Figure 3(c). The input vector and the filter have a finite number of non-zero entries indicated with yellow and orange squares/rectangles, respectively, and the output has a finite number of non-zero entries indicated with red squares/rectangles. For simplicity, we identify the infinite vectors in with vectors in where is the number of their non-zero entries. Given the illustrative purpose of these examples, for simplicity we ignore boundary effects. The signals’ sizes are:
, input vector , output vector ;
, input vector , output vector ;
, input vector , output vector .
When the stride is an integer, equation (19) describes what is represented in Figures 3(a) and 3(b). When the stride is , as depicted in Figure 3(c), it is intuitive to consider a filter whose entries are half the size of the input ones. This is equivalent to choosing the filter in a space of higher resolution with respect to the space of the input signal. As a result, the output belongs to the same higher resolution space. For instance, the filter belongs to a space that is twice the resolution of the input space when the stride is . This notion of resolution is coherent with the scale parameter used in the continuous setting.
a.3 Proof of Theorem 1
We begin with some preliminary technical lemmas.
Let Hypothesis 1 hold and let
where is the Fourier series of , defined as