Emergence of hierarchical modes from deep learning

Introduction.— Deep neural networks are dominant tools with a broad range of applications in not only image and language processing, but also scientific researches Goodfellow et al. (2016); Carleo et al. (2019). These networks are parameterized by a huge amount of trainable weight matrices, thereby consuming an expensive training cost. However, these weight matrices are hard to interpret, and thus mechanisms underlying the macroscopic performance of the networks remain a big mystery in theoretical studies of neural networks Huang (2022); Roberts et al. (2022).

To save the computational cost, previous studies of deep networks applied singular value decomposition to the weight matrices Jaderberg et al. (2014); Yang et al. (2020); Giambagli et al. (2021); Chicchi et al. (2021). This decomposition requires the orthogonality condition for the singular vectors and positive singular values. The training also involves a carefully-designed structure for the trainable decomposition scheme Giambagli et al. (2021); Chicchi et al. (2021). These constraints and designs make the training process complicated, and thus a concise physics interpretation is still lacking. In addition, previous studies of recurrent memory networks showed that the network weight can be decomposed into separate random orthogonal patterns with corresponding importance scores Jiang et al. (2021); Zhou et al. (2021). Inspired by these studies, we conjecture that the learning in deep networks is shaped by a hierarchy of latent modes, which are not necessarily orthogonal, and the weight matrix can be expressed by these modes.

The mode decomposition learning (MDL) leads to a progressively compact latent mode space across the network hierarchy, and meanwhile the subspaces corresponding to different types of input are strongly disentangled, facilitating discrimination. The least number of latent modes achieving the comparable performance with the costly standard methods grows only logarithmically with the network width and even could be a constant, thereby reducing significantly the training cost. The mode spectrum exhibits an intriguing piecewise power-law behavior. In particular, these properties do not depend on details of the training setting. Therefore, our proposed MDL calls for a rethinking of conventional weight-based deep learning through the lens of cheap and interpretable mode-based learning.

Model.— To show the effectiveness of the MDL scheme, we train a deep network to implement a classification task of handwritten digits mni . The deep network has $L$ layers ($L-2$ hidden layers) with $N_{l}$ neurons in the $l$-th layer. The weight value of the connection from the neuron $i$ at the upstream layer $l$ to the neuron $j$ at the downstream layer $l+1$ is specified by $w_{ij}^{l}$. The activation of the neuron $j$ at the downstream layer $h_{j}^{l+1}=f(z_{j}^{l+1})=\max(0,z_{j}^{l+1})$, where the pre-activation $z_{j}^{l+1}=\sum_{i}w_{ij}^{l}h_{i}^{l}$. For the output layer, the softmax function $h_{k}=e^{z_{k}}/\sum_{i}e^{z_{i}}$ is chosen to specify the probability over all classes of the input images. The cross entropy $\mathcal{C}=-\sum_{i}\hat{h}_{i}\ln h_{i}$ is used as the cost function for the supervised learning, and $\hat{h}_{i}$ is the target label (one-hot form). After training (the cross entropy is repeatedly averaged over mini-batches of training examples), we evaluate the generalization performance of the network on an unseen test dataset.

Single weight values are not interpretable. According to our hypothesis, latent patterns would emerge from training in each layer. We call these patterns hierarchical modes for deep learning. Therefore, the relationship between the modes and weight values is expressed by the following mode decomposition,

where there are $p^{l}$ upstream modes $\bm{\hat{\xi}}^{l}\in\mathbb{R}^{N_{l}\times p^{l}}$, and the same number of downstream modes $\bm{\xi}^{l+1}\in\mathbb{R}^{N_{l+1}\times p^{l}}$. The importance of each pair of adjacent modes is specified by the diagonal of the importance matrix $\bm{\Sigma}^{l}\in\mathbb{R}^{p^{l}\times p^{l}}$. These modes may not be orthogonal with each other, and the importance score can take a real value. This setting allows for more degrees of freedom for learning features of input-output mappings. We will detail their geometric and physical interpretations below.

A geometric interpretation of Eq. (1) in a simple learning task is shown in Fig. 1. We use a three-layer network with three hidden neurons. The input data is sampled from a four-component Gaussian mixture Fischer et al. (2022),

where $\mathcal{N}\left(\mathbf{x}|\mu_{x}^{t,\pm},\Sigma_{x}^{t,\pm}\right)$ denotes a Gaussian distribution with mean $\mu_{x}^{t,\pm}$ and covariances $\Sigma_{x}^{t,\pm}$, and $P(t)=P_{\pm}=\frac{1}{2}$. For the label $t=+1$, $\mu_{x}^{t=+1,\pm}=\pm(0.5,0.5)^{\operatorname{T}}$, while for $t=-1$, $\mu_{x}^{t=-1,\pm}=\pm(-0.5,0.5)^{\operatorname{T}}$. Covariances are isotropic throughout with $\Sigma_{x}^{t,\pm}=0.05\mathds{1}$. The input samples $\mathbf{x}\in\mathbb{R}^{2}$ are first projected to the input pattern space spanned by $(\hat{\bm{\xi}}^{1})^{\operatorname{T}}_{i}$ ($i=1,2,3$). Then all three directions of this projection get expanded or contracted via $\bm{\Sigma}^{1}(\bm{\hat{\xi}}^{1})^{\operatorname{T}}\mathbf{x}$. Finally the geometrically modified representation is re-mapped to the downstream representation space of a higher dimensionality, as $\bm{{\xi}}^{2}\bm{\Sigma}^{1}(\bm{\hat{\xi}}^{1})^{\operatorname{T}}\mathbf{x}$ [Fig. 1 (c)]. The non-linearity of the transfer function is then applied to the last linear transformation, leading to the geometric separation [Fig. 1 (b)]. We conclude that the MDL provides rich angles to look at the geometric transformation of the input information along the hierarchy of deep networks.

Rather than the conventional weight values in standard backpropagation (BP) algorithms Goodfellow et al. (2016), the trainable parameters are latent patterns in the MDL. The training is implemented by stochastic gradient descent in the mode space $\bm{\theta}^{l}=(\bm{\hat{\xi}}^{l},\bm{\Sigma}^{l},\bm{{\xi}}^{l+1})$ SM ,

where $\mathcal{L}$ denotes the cost function (e.g., cross-entropy or mean-squared error) over a mini-batch of training data, $\eta$ denotes the learning rate, and $\mathcal{K}_{j}^{l+1}\equiv\partial\mathcal{L}/\partial z_{j}^{l+1}$ denotes the error term, which could back-propagate from the top layer where $\mathcal{K}_{j}^{L}=-\hat{h}_{j}^{L}\left(1-h_{j}^{L}\right)$ for $\mathcal{L}=\mathcal{C}$ (cross entropy). Based on the chain rule, the error backpropagation equation can be derived as $\mathcal{K}_{i}^{l}=\sum_{j}\mathcal{K}_{j}^{l+1}\sum_{\alpha}\xi_{i\alpha}^{l+1}\Sigma_{\alpha}^{l}\hat{\xi}_{j\alpha}^{l}f^{\prime}(z_{i}^{l})$ SM . To ensure the pre-activation is independent of the upstream-layer width, we take the initialization scheme that $[\bm{{\xi}}^{l+1}\bm{{\Sigma}}^{l}(\hat{\bm{\xi}}^{l})^{\operatorname{T}}]_{ij}\sim\mathcal{O}(\frac{1}{\sqrt{N_{l}}})$ Jiang et al. (2021). To avoid the ambiguity of choosing patterns (e.g., scaled by a factor), we impose an identical regularization with strength $10^{-4}$ for all trainable parameters. However, our result does not change qualitatively with the specific values of regularization SM .

We remark that for each hidden layer, there exist two types of pattern ($\bm{\xi}^{l}\neq\hat{\bm{\xi}}^{l}$). Equation (3) is used to learn these patterns. We call this case 1L2P. If we assume $\bm{\xi}^{l}=\hat{\bm{\xi}}^{l}$, the training can be further simplified as in SM , and we call this case 1L1P. The nature of this mode-based-computation can be understood as an expanded linear-nonlinear layered computation, as $f(z_{j}^{l+1})=f(\sum_{\alpha}c_{\alpha j}\kappa_{\alpha})$ where the linear field $\kappa_{\alpha}=\sum_{i}\hat{\xi}_{i\alpha}^{l}h_{i}^{l}$ and the equivalent weight $c_{\alpha j}=\xi_{j\alpha}^{l+1}\Sigma_{\alpha}^{l}$. Therefore, the number of modes acts as the linear-layer width. We leave a systematic exploration of this linear-nonlinear structure by statistical mechanics in forthcoming works.

On-line learning dynamics in a shallow network.— The MDL can be analytically understood in an on-line learning setting, where we consider one-hidden-layer architecture. The on-line learning can be considered as a special case of the above mini-batch learning (i.e., the batch size is set to one, and the sample is visited by the learning only once). The training dataset consists of $n$ pairs $\left\{\mathbf{x}^{\nu},y^{\nu}\right\}_{\nu=1}^{n}$. Each training example is independently sampled from a probability distribution $\mathbb{P}(\mathbf{x},y)=\mathbb{P}(y|\mathbf{x})\mathbb{P}(\mathbf{x})$, where $\mathbb{P}(\mathbf{x})$ is a standard Gaussian distribution, and the scalar label ${y}^{\nu}$ is generated by the neural network of $k$ hidden neurons, (i.e., teacher, indicated by the symbol $*$ below). Given an input $\mathbf{x}^{\nu}\in\mathbb{R}^{d}$, the corresponding label is created by

where $[\bm{\xi}^{*}\bm{\Sigma}^{*}(\hat{\bm{\xi}}^{*})^{\operatorname{T}}]_{r}$ denotes the $r$-th row of the matrix $\bm{\xi}^{*}\bm{\Sigma}^{*}(\hat{\bm{\xi}}^{*})^{\operatorname{T}}$, and $\lambda_{r}^{*\nu}=[\bm{{\xi}}^{*}\bm{{\Sigma}^{*}}(\bm{\hat{\xi}^{*}})^{\operatorname{T}}]_{r}\mathbf{x}^{\nu}/{\sqrt{d}}$ represents the $r$-th element of the teacher local field vector $\bm{\lambda}^{*\nu}\in\mathbb{R}^{k}$. The teacher network is quenched as $[\bm{{\xi}}^{*}\bm{{\Sigma}^{*}}(\bm{\hat{\xi}^{*}})^{\operatorname{T}}]_{ij}\sim\mathcal{O}(1)$. Here, we focus on the non-linear transfer function $\sigma(x)=\operatorname{erf}(x/\sqrt{2})$. In addition, we train the other shallow network called the student network, by minimizing the loss function $\mathcal{L}(y,\hat{f}(\mathbf{x},\Theta))$ over the training data (labels are given by the teacher network), where $\Theta$ denotes the trainable parameters. The student’s prediction for a fresh sample $\mathbf{x}$ is given by

where $\lambda_{r}$ denotes the $r$-th component of the student local field $\bm{\lambda}=\bm{{\xi}}\bm{{\Sigma}}(\bm{\hat{\xi}})^{\top}\mathbf{x}$, and the student has $m$ hidden neurons. The student is supplied with data samples in sequence (one sample each time step). We next use $\nu$ to indicate the time step as well.

The mean-squared-error can be evaluated as

where $f(\cdot)$ indicates the teacher’s output, and we have replaced the expectation $\mathbb{E}_{\mathbf{x},y\sim\mathbb{P}(\mathbf{x},y)}[\cdot]$ by $\mathbb{E}_{\bm{\lambda},\bm{\lambda}^{*}\sim\mathcal{N}\left(\bm{\lambda},\bm{\lambda}^{*}\mid 0,\bm{\Omega}\right)}[\cdot]$, because of the central-limit theorem and the i.i.d. setting we consider Biehl and Schwarze (1995); Saad and Solla (1995); Goldt et al. (2019). The covariance of the local field $\bm{\Omega}^{\nu}\in\mathbb{R}^{(k+m)\times(k+m)}$ can be specified as follows,

where $\mathbf{Q}^{\nu}\equiv\mathbb{E}_{\mathbf{x},y\sim\mathbb{P}(\mathbf{x},y)}\left[\bm{\lambda}^{\nu}(\bm{\lambda}^{\nu})^{\operatorname{T}}\right]$, $\mathbf{M}^{\nu}\equiv\mathbb{E}_{\mathbf{x},y\sim\mathbb{P}(\mathbf{x},y)}\left[\bm{\lambda}^{\nu}(\bm{\lambda}^{*\nu})^{\operatorname{T}}\right]$, and $\mathbf{P}^{\nu}\equiv\mathbb{E}_{\mathbf{x},y\sim\mathbb{P}(\mathbf{x},y)}\left[\bm{\lambda}^{*\nu}(\bm{\lambda}^{*\nu})^{\operatorname{T}}\right]$. By definition, $\mathbf{P}$ is fixed, while $\mathbf{Q}^{\nu}$ and $\mathbf{M}^{\nu}$ evolve according to the gradient updates, following a set of deterministic ordinary differential equations (ODEs) as the input dimension $d\to\infty$ SM . These matrices are exactly the order parameters in physics. For simplicity, we consider $\bm{\xi}=\bm{\xi}^{*}$ and $\bm{\Sigma}=\bm{\Sigma}^{*}$, i.e., only the upstream patterns are learned.

Results.— MDL can reach a similar test accuracy with that of BP performed in the weight space, when $p$ is sufficiently large [Fig. 2 (a)]. The computational cost of the BP scales with $N_{l}^{2}$. In contrast, MDL works in the mode space, requiring a training cost of only the order of $pN_{l}$. Note that $p$ is much smaller than $N_{l}$ (or $\lim_{N_{l}\to\infty}p^{l}/N_{l}=0$), and our MDL does not need any additional training constraints (compared to other matrix factorization algorithms SM ). Remarkably, when $p=30$, the performance of MDL already matches that of BP [Fig. 2 (b)], but only utilizes $40\%$ of the full sets of parameters that are consumed by the BP. In fact, each hidden layer can have two different types of latent pattern (1L2P) due to the mode decomposition. But if we assume that $\bm{{\xi}}^{l}=\bm{\hat{\xi}}^{l}$, i.e., each layer share a single type of pattern (1L1P), we can further reduce the computational cost by an amount of $\sum_{l}p^{l}N_{l}$, without sacrificing the test accuracy [Fig. 2 (b)]. Varying the network width, we reveal a logarithmic increase of the least number of modes [Fig. 2 (c)], which is a novel property of deep learning in the mode space, in stark contrast to a linear number of memory patterns in previous studies Jiang et al. (2021). When the network width further grows, the least number can even become a constant. We argue that this manifests three separated phases of poor-good-saturated performance with increasing layer width (see Fig. S9 in SM ).

To see how the latent patterns are transformed in geometry along the network hierarchy, we first calculate the center of the pattern space. Then the Euclidean distance from this center to each pattern is analyzed. We find that the pattern space becomes progressively compact when going to deep layers [Fig. 2 (d)]. To further characterize the geometric details, we define the subspace spanned by the principal eigenvectors of the layer neural responses to one type of inputs. Then the subspace overlap is calculated as the cosine of the principal angle between two subspaces corresponding to two types of inputs Bjoerck and Golub (1973); SM . We find that the hidden-layer representation becomes more disentangled with layer in comparison with BP [Fig. 2 (e,f)]. MDL shows great computational benefits of representation disentanglement, thereby facilitating discrimination. A slight increase of the overlap is observed for deeper layers, which is caused by the saturation of the test performance (see more analyses in SM ).

Compared to other matrix factorization methods, MDL has no additional constraints for the modes and importance scores, therefore being flexible for feature extraction. We find that the interlayer patterns are less orthogonal than the intralayer ones. The geometric transformation carried out by these latent pattern matrices is not strictly a rotation for which the $\ell_{2}$ norm is preserved. This flexibility may be the key to make our method better than other matrix factorization methods in both training cost and learning performance (see details in SM ).

We next ask whether some modes are more important than the others. Therefore, we rank the modes according to the measure $\tau_{\alpha}=\gamma\|\xi_{\alpha}\|_{2}+\gamma\|\hat{\xi}_{\alpha}\|_{2}+|\Sigma_{\alpha}|$, where $\gamma=\sum_{\alpha}|{\Sigma_{\alpha}}|/\sum_{\alpha}(\|{\xi}_{\alpha}\|_{2}+\|{\hat{\xi}}_{\alpha}\|_{2})$ to make comparable the magnitudes of the pattern and importance ($\bm{\Sigma}$) score. Removing modes with weak values of $\tau$ first yields much higher accuracy than the random removal protocol [Fig. 3 (a)], suggesting the existence of leading modes. Moreover, deeper layers are more robust. Figure 3 (b) shows the measure as a function of rank in descending order, which can be approximately captured by piecewise power-law behavior (a transition point at the rank $10$). Ranking with only the importance scores yields similar behavior SM . A small exponent is observed for the leading measures, while the remaining measures bear a large exponent, thereby revealing the coding hierarchy of latent modes in the deep networks. This intriguing behavior does not change with the regularization strength or the hidden-layer width SM .

Finally, the on-line mean-squared error dynamics of our model can be predicted perfectly in a teacher-student setting. The number of modes strongly affects the shape of the learning dynamics, and a large mode load can make the plateaus disappear (Fig. 4). Moreover, during learning, the alignment between receptive fields of the student’s hidden nodes and the teacher’s ones continuously emerge, which is called the specialization transition Schwarze (1993); Goldt et al. (2019).

Conclusion.— In this Letter, we propose a mode decomposition learning that works in the mode space rather than the conventional weight space. This learning scheme has three-fold technical and conceptual advances. First, the learning can achieve the comparable performance with standard methods, with a significant reduction of training costs. We also find that the least number of modes grows only logarithmically with the network width and becomes even independent of larger width, which is in stark contrast to a linear number of patterns in recurrent memory networks. Second, the learning leads to progressively compact pattern spaces, which promotes highly disentangled hierarchical representations. The upstream pattern maps the activity into a low-dimensional space, and then the resulting embedding is further expanded or contracted. After that, the modified embedding is re-mapped into the high-dimensional activity space. This sequence of geometric transformation can be understood as a linear-nonlinear hidden structure. Third, all modes are not equally important to the generalization ability of the network, showing an intriguing piecewise power-law behavior. Finally, the mode learning dynamics can be predicted by the mean-field ODEs, revealing the mode specialization transition. Therefore, the MDL inspires a rethinking of conventional deep learning, offering a faster, more interpretable training framework. Future works along this direction will be inspired. For example, the impact of other structured dataset, mode dynamics in over-parameterized or recurrent networks, and the origin of adversarial vulnerability of deep networks in terms of geometry of the mode space.