Associative memory and dead neurons

## Abstract

Abstract In “Large Associative Memory Problem in Neurobiology and Machine Learning,” Dmitry Krotov and John Hopfield introduced a general technique for the systematic construction of neural ordinary differential equations with non-increasing energy or Lyapunov function. We study this energy function and identify that it is vulnerable to the problem of dead neurons. Each point in the state space where the neuron dies is contained in a non-compact region with constant energy. In these flat regions, energy function alone does not completely determine all degrees of freedom and, as a consequence, can not be used to analyze stability or find steady states or basins of attraction. We perform a direct analysis of the dynamical system and show how to resolve problems caused by flat directions corresponding to dead neurons: (i) all information about the state vector at a fixed point can be extracted from the energy and Hessian matrix (of Lagrange function), (ii) it is enough to analyze stability in the range of Hessian matrix, (iii) if steady state touching flat region is stable the whole flat region is the basin of attraction. The analysis of the Hessian matrix can be complicated for realistic architectures, so we show that for a slightly altered dynamical system (with the same structure of steady states), one can derive a diverse family of Lyapunov functions that do not have flat regions corresponding to dead neurons. In addition, these energy functions allow one to use Lagrange functions with Hessian matrices that are not necessarily positive definite and even consider architectures with non-symmetric feedforward and feedback connections.

## 1 Introduction

Associative or content-addressable memory is a system that retrieves the most appropriate stored pattern based on a partially known or distorted input pattern. One particularly influential realization of associative memory was proposed by John Hopfield in [] for discrete variables and in [] for continuous variables. Both models are distinguished by their biological plausibility, autonomy, asynchronous operations of constituent parts, robustness to noise, and strong theoretical guarantees. Later in [], it was shown that one could develop a general biologically plausible model that unites many previously known models and allows building novel associative memory systems [].

The model in [] is based on the nonlinear dynamical system that evolves in time from a given initial state. Nonlinear dynamical systems show an exceptionally diverse set of behavior [], so one needs to select an appropriate class of ordinary differential equations suitable to model associative memory. The most cluical requirement is the ability of the system to evolve to a single state from many initial conditions that are close according to some problem-specific metrics. This fact suggests one should use a dynamical system with many stable steady states. If one selects such a system, each stable steady state corresponds to a particular memory and basin of attraction – all initial conditions that evolve to a selected state – defines a measure of similarity between states.

The technique of choice to study the stability of steady state is to construct energy or Lyapunov function []. This function, defined on the state of a dynamical system, is non-increasing on trajectories of a dynamical system. If it is possible to find such a function, its isolated local minima will correspond to steady states. Different variants of energy functions with this property are available in all previous works on associative memory [], [], [] and recent work [] provides the most general energy function that embraces all previous ones.

Since associative memory from [] has an associated energy function, one can argue that it is a member of a broad family of energy-based models [], including the diffusion models []. In addition to that, the dynamical system used is clearly related to another subfield of machine learning known as Neural Ordinary Differential Equation []. Finally, in [], [] it was shown that one can select parameters of model [] in such a way that dynamical variables correspond to the activation pattern of deep neural network with feedback connection. Given that, one can argue that modern Hopfield associative memory [] is uniquely tying up several major paradigms in deep learning.

In this article, we study the Lyapunov function proposed in [] and later adopted and modified in other recent papers, e.g., [], [], []. Our main observation is that energy function is vulnerable to the problem of dead neurons [], which result in a flat energy direction. We illustrate this in the left panel of Figure 1, where one can find both the vector field of the dynamical system and isolines of the energy function. Clearly, the energy function from [] does not ensure stability and is not helpful in identifying a steady state when dead neurons are present.

As a remedy, we propose a slightly modified dynamical system that has no flat direction (see right panel of Figure 1) but retains other good properties of the Krotov and Hopfield model. In particular, it is still an energy-based model, neural ODE, and has steady states related to deep neural networks.

Figure: (a) Energy function from [].
Refer to caption: x1.png

A more detailed breakdown of our contribution is below:

- 1.
In Section 2 we provide examples of flat energy directions caused by dead neurons (Figure 1; Examples 2, 3, 4) and formally characterize architectures that are vulnerable to that problem in Proposition 1.
- 2.
Flat energy directions, as we show in Section 3, cause several undesirable consequences for sensitivity and stability:
(a)
Problems with sensitivity described in Section 3.1 include: (i) energy is completely independent of degrees of freedom corresponding to dead neurons, because of that activation functions are effectively invertible and energy function presented in [] is completely equivalent to the old one given in []; (ii) In the flat regions energy function is not sensitive to the change in bias term (Section 3.1).
(b)
Stability is discussed in Section 3.2 with the following main takeaways: (i) for steady state with dead neurons, stability condition can not be ensured from the Lyapunov function alone; (ii) independent analysis of dynamical system shows dead neurons do not compromise stability, and only the motion in the range of Hessian of Lagrange function is relevant; (iii) if the steady state is stable, the whole flat direction belongs to the basin of attraction; (iv) if in addition of energy, the range of the Hessian is available (e.g., as the orthogonal projector), this information is enough to restore degrees of freedom that energy alone misses at steady state.
- 3.
As a remedy, in Section 4, we define a slightly modified dynamical system and a family of energy functions with good properties: (i) proposition 5 shows that in general one may construct a large family of Lyapunov function with no flat directions; (ii) example 5 shows several concrete choices of Lyapunov functions for symmetric weight matrices with no restriction Hessian matrix of Lagrange function; (iii) examples 6 and 7 further relax restriction on parameters and allow one to consider memory models with non-symmetric weight matrices.

## 2 Dead neurons

In [] Krotov and Hopfield proposed the following set of equations to model associative memory

$$ $\displaystyle\dot{\boldsymbol{y}}(t)=\boldsymbol{W}\boldsymbol{g}(\boldsymbol{ y}(t))-\boldsymbol{y}(t)+\boldsymbol{b},\,\boldsymbol{y}(0)=\boldsymbol{y}_{0},$ (1) $\displaystyle\boldsymbol{g}(\boldsymbol{y})=\frac{\partial L(\boldsymbol{y})}{ \partial\boldsymbol{y}},\,\boldsymbol{W}=\boldsymbol{W}^{\top},$ (2) $\displaystyle\Lambda_{ij}(\boldsymbol{y})=\frac{\partial^{2}L(\boldsymbol{y})} {\partial y_{i}\partial y_{j}},\,\boldsymbol{\Lambda}(\boldsymbol{y})\geq 0,$ (3) $\displaystyle E(\boldsymbol{y})=\left(\boldsymbol{y}-\boldsymbol{b}\right)^{ \top}\boldsymbol{g}(\boldsymbol{y})-L(\boldsymbol{y})-\frac{1}{2}\left( \boldsymbol{g}(\boldsymbol{y})\right)^{\top}\boldsymbol{W}\boldsymbol{g}( \boldsymbol{y}).$ (4) $$

Equation (1) describes a temporal dynamics of feature vector $\boldsymbol{y}$ starting from $\boldsymbol{y}_{0}$. The equation contains weights $\boldsymbol{W}$, bias $\boldsymbol{b}$ and activation function $\boldsymbol{g}(\boldsymbol{y})$. Structural assumptions (2) ensure that the energy function (4) exists, and restriction on the Hessian $\boldsymbol{\Lambda}$ of Lagrange function $L(\boldsymbol{y})$ (3) guarantees that energy (4) is non-increasing on trajectory of dynamical system (1).

Dynamical system (1) has several favorable properties: (i) steady states can be used as memory vectors, (ii) dynamics in the basins of attraction naturally model memory recovery process, (iii) energy function can be used to ensure the existence of steady states and basins of attraction, (iv) memory is related to neural ODEs so can be trained end-to-end, (v) with a special choice of $\boldsymbol{W}$ steady states resemble activation pattern of classical deep learning architectures. The example below illustrates the last point.

###### Example 1 (MLP with feedback connections) .

For simplicity, we consider four layers, extensions to a larger number of layers are straightforward. Weights and state vectors are partitioned on blocks of conformable size

$$ $\boldsymbol{W}=\begin{pmatrix}0&\boldsymbol{W}_{12}&\\ \boldsymbol{W}_{21}&0&\boldsymbol{W}_{23}\\ &\boldsymbol{W}_{32}&0&\boldsymbol{W}_{34}\\ &&\boldsymbol{W}_{43}&0\end{pmatrix},\boldsymbol{y}=\begin{pmatrix}\boldsymbol {y}_{1}\\ \boldsymbol{y}_{2}\\ \boldsymbol{y}_{3}\\ \boldsymbol{y}_{4}\end{pmatrix},\,\boldsymbol{b}=\begin{pmatrix}\boldsymbol{b} _{1}\\ \boldsymbol{b}_{2}\\ \boldsymbol{b}_{3}\\ \boldsymbol{b}_{4}\end{pmatrix}.$ $$

Note that $\boldsymbol{W}$ is symmetric so $\boldsymbol{W}_{12}=\boldsymbol{W}_{2}1^{\top}$
With that choice dynamical system (1) becomes

$$ $\dot{\boldsymbol{y}}_{i}=\boldsymbol{W}_{ii-1}\boldsymbol{g}\left(\boldsymbol{ y}_{i-1}\right)+\boldsymbol{W}_{ii+1}\boldsymbol{g}\left(\boldsymbol{y}_{i+1} \right)-\boldsymbol{y}_{i}+\boldsymbol{b}_{i},$ $$

so steady-state indeed resembles MLP but with feedback connections that are symmetric.

Besides MLP described in Example 1 associative memory (1) leads to many more interesting and fruitful connections. In [] it allowed authors to reconstruct dense associative memory [] and modern Hopfield network []. In [] and [] it was used to build a memory model with dense hidden layers and convolution neural networks with pooling layers. In [] authors introduced an energy transformer using the same formalism. Authors of [] also noticed that MLP-mixer [] is related to associative memory (1). Clearly, the technique is valuable and versatile.

As we have seen, steady states of dynamical systems are important to emulate deep learning architectures and classical memory models. The role of the Lyapunov function is to ensure stability. Unfortunately, if one looks closer at the energy function (4), it becomes clear that it shows some pathological behavior. Before providing a formal description, we illustrate this with three examples.

###### Example 2 (flat energy with ReLU activations) .

Lagrange function $L(\boldsymbol{y})=\sum_{i=1}^{N}\frac{1}{2}\left(\text{ReLU}(y_{i})\right)^{2}$ corresponds to fully-connected neural network with $N$ neurons and $\text{ReLU}(x)=\frac{1}{2}\left(x+|x|\right)$ nonlinearity. Energy function becomes $E(\boldsymbol{y})=\left(\boldsymbol{y}-\boldsymbol{b}\right)^{\top}\text{ReLU}
(\boldsymbol{y})-\sum_{i=1}^{N}\frac{1}{2}\left(\text{ReLU}(y_{i})\right)^{2}-
\frac{1}{2}\left(\text{ReLU}(\boldsymbol{y})\right)^{\top}\boldsymbol{W}\,
\text{ReLU}(\boldsymbol{y})$. It is easy to see that if neuron $i$ dies, i.e., $y_{i}\leq 0$, energy becomes zero for all $\widetilde{\boldsymbol{y}}=\boldsymbol{y}+\alpha\boldsymbol{e}_{i}$, where $\alpha\leq 0$. $\boldsymbol{e}_{i}$ is a $i$-th columns of identity matrix and $\boldsymbol{y}$ is a state vector with $y_{i}\leq 0$. In other words, energy is non-discriminative in a non-compact region of state space.

###### Example 3 (flat energy with sigmoid activations) .

For sigmoid activation function $\sigma(x)=\left(1+\exp(-x)\right)^{-1}$ Lagrange function reads $L(\boldsymbol{y})=\sum_{i}\log\left(1+e^{y_{i}}\right)$, and the energy is $E(\boldsymbol{y})=\left(\boldsymbol{y}-\boldsymbol{b}\right)^{\top}\boldsymbol
{\sigma}(\boldsymbol{y})-\sum_{i}\log\left(1+e^{y_{i}}\right)-\frac{1}{2}\left
(\boldsymbol{\sigma}(\boldsymbol{y})\right)^{\top}\boldsymbol{W}\,\boldsymbol{
\sigma}(\boldsymbol{y})$. Neuron $i$ dies when sigmoid saturates for some $y_{i}\gg 1$. After that for all $\widetilde{\boldsymbol{y}}=\boldsymbol{y}+\alpha\boldsymbol{e}_{i}$, with $\alpha\geq 0$ the energy has flat region, i.e., $E(\boldsymbol{y})=E(\widetilde{\boldsymbol{y}}(\alpha))$ for all admissible $\alpha$.

###### Example 4 (flat energy with softmax activations) .

Activation $\text{softmax}\left(\boldsymbol{y}\right)=\exp(\boldsymbol{y})\big{/}\sum_{i=1
}^{N}\exp(y_{i})$ corresponds to $L(\boldsymbol{y})=\log\left(\sum_{i=1}^{N}\exp(y_{i})\right)$ and energy $E(\boldsymbol{y})=\left(\boldsymbol{y}-\boldsymbol{b}\right)^{\top}\text{
softmax}\left(\boldsymbol{y}\right)-\log\left(\sum_{i=1}^{N}\exp(y_{i})\right)
-\frac{1}{2}\left(\text{softmax}\left(\boldsymbol{y}\right)\right)^{\top}
\boldsymbol{W}\,\text{softmax}\left(\boldsymbol{y}\right)$. Consider variable $\widetilde{\boldsymbol{y}}=\boldsymbol{y}+\boldsymbol{c}$, where $\boldsymbol{c}$ is a vector with all components equal $c\in\mathbb{R}$. So constant shift, $\text{softmax}\left(\boldsymbol{y}\right)=\text{softmax}\left(\widetilde{
\boldsymbol{y}}\right)$, Lagrange function shifts on $c$ and $\widetilde{\boldsymbol{y}}^{\top}\text{softmax}(\widetilde{\boldsymbol{y}})=
\boldsymbol{y}^{\top}\text{softmax}(\boldsymbol{y})+c$, so overall energy remains the same, i.e., $E(\boldsymbol{y})=E(\widetilde{\boldsymbol{y}}(c))$ for arbitrary $c$.

The examples above demonstrate that energy function (4) fails to distinguish states on a large fraction of state space. In Examples 2 and 3 this happens because the activation function saturates for some inputs, and in Example 4 this is a consequence of invariance. The latter case is typically not considered as a dead neuron, but since the number of degrees of freedom decreases by one we call this “effective neuron” dead all the same. The formal definition of dead neurons appears below (see [], [], [] for the discussion of dead neurons in the context of ReLU activation function).

###### Definition (dead neurons) .

At a given point $\boldsymbol{y}\in\mathbb{R}^{N}$ activation function $\boldsymbol{g}$ has $k$ dead neurons if it is possible to find $\boldsymbol{V}\in\mathbb{R}^{N\times k}$ such that $\boldsymbol{g}\left(\boldsymbol{y}+\boldsymbol{V}\boldsymbol{c}\right)=
\boldsymbol{g}\left(\boldsymbol{y}\right)$ for all $\boldsymbol{c}\in\mathbb{R}^{k}_{+}$.

The examples above hint that most of the activation functions will lead to dead neurons. The first large class is activation functions with saturation: hyperbolic tangent, sigmoid, ReLU, GELU, SiLU, SELU, Gaussian, etc. The second class is activation functions with symmetries, e.g., softmax and layer norm.

With the definition of dead neurons, we can formalize a pathological behavior of energy function (4).

###### Proposition 1 .

If at a given point $\boldsymbol{y}$ activation function $\boldsymbol{g}$ has $k$ dead neuron defined by $\boldsymbol{V}\in\mathbb{R}^{N\times k}$, energy function (4) has constant value in a subspace $\mathcal{D}=\left\{\boldsymbol{y}+\boldsymbol{V}c:\boldsymbol{c}\in\mathbb{R}^
{k}_{+}\right\}$.

###### Proof.

Taking $\widetilde{\boldsymbol{y}}=\boldsymbol{y}+\boldsymbol{V}\boldsymbol{c}$, we find from Taylor’s theorem with mean-value form of reminder $L(\widetilde{\boldsymbol{y}})=L(\boldsymbol{y})+\boldsymbol{g}(\boldsymbol{y}+
\tau\boldsymbol{V}\boldsymbol{c})^{\top}\boldsymbol{V}\boldsymbol{c}$ for some $\tau\in(0,1)$. Since $\boldsymbol{g}(\boldsymbol{y}+\tau\boldsymbol{V}\boldsymbol{c})=\boldsymbol{g}
(\boldsymbol{y})$ by assumption, the second term in the energy function (4) shifts by $-\boldsymbol{g}(\boldsymbol{y})^{\top}\boldsymbol{V}\boldsymbol{c}$ which compensates similar change in the first term, so overall energy does not change under the transformation $\boldsymbol{y}\rightarrow\boldsymbol{y}+\boldsymbol{V}\boldsymbol{c}$.
∎

Flat energy directions are illustrated in Figure 1. Intuitively, it is clear that having large regions with flat energy is not good. In the next section, we explain in detail what will go awry.

###### Example 1 (MLP with feedback connections) .

###### Example 2 (flat energy with ReLU activations) .

###### Example 3 (flat energy with sigmoid activations) .

###### Example 4 (flat energy with softmax activations) .

###### Definition (dead neurons) .

###### Proposition 1 .

###### Proof.

## 3 Consequences of dead neurons

Proposition 1 implies that for most architectures used in practice, there are large regions of state space with flat energy functions. This has several negative consequences that we somewhat arbitrarily classify as problems with sensitivity and stability.

### 3.1 Sensitivity

As evident from Figure 1 energy is flat in directions $\boldsymbol{V}$ corresponding to dead neurons. This can be formalised if one considers new variables $\boldsymbol{y}_{d}=\boldsymbol{V}\boldsymbol{V}^{\top}\boldsymbol{y}$, $\boldsymbol{y}_{a}=\left(\boldsymbol{I}-\boldsymbol{V}\boldsymbol{V}^{\top}
\right)\boldsymbol{y}$ corresponding to dead and alive neurons. Since $\boldsymbol{y}=\boldsymbol{y}_{d}+\boldsymbol{y}_{a}$, the energy (4) becomes

$$ $E(\boldsymbol{y}_{d},\boldsymbol{y}_{a})=\left(\boldsymbol{y}_{d}+\boldsymbol{ y}_{a}-\boldsymbol{b}\right)^{\top}\boldsymbol{g}(\boldsymbol{y}_{a})-L( \boldsymbol{y}_{d}+\boldsymbol{y}_{a})-\frac{1}{2}\left(\boldsymbol{g}( \boldsymbol{y}_{a})\right)^{\top}\boldsymbol{W}\boldsymbol{g}(\boldsymbol{y}_{ a})\\ =\left(\boldsymbol{y}_{a}-\boldsymbol{b}\right)^{\top}\boldsymbol{g}( \boldsymbol{y}_{a})-L(\boldsymbol{y}_{a})-\frac{1}{2}\left(\boldsymbol{g}( \boldsymbol{y}_{a})\right)^{\top}\boldsymbol{W}\boldsymbol{g}(\boldsymbol{y}_{ a}),$ $$

where we used $\boldsymbol{g}(\boldsymbol{y}_{d}+\boldsymbol{y}_{a})=\boldsymbol{g}(
\boldsymbol{y}_{a})$ and $L(\boldsymbol{y}_{d}+\boldsymbol{y}_{a})=L(\boldsymbol{y}_{a})+\boldsymbol{g}(
\boldsymbol{y}_{a})^{\top}\boldsymbol{y}_{d}$.

We see that energy does not depend on variables $\boldsymbol{y}_{d}$ corresponding to dead neurons, which means the effective number of degrees of freedom is decreased from $N$ to $N-k$ where $k$ is the number of dead neurons. If only energy is available it is not possible to recover values on $\boldsymbol{y}_{d}$ and steady states can not be found as in examples from Figure 1.

Variables $\boldsymbol{y}_{d}$ are from the kernel of $\boldsymbol{g}$ which means the activation function becomes effectively invertible. In [], the energy function for this situation has already been introduced. It is instructive to compare a new energy function [] with the old one. Lyapunov function from the 1984 paper, in the notation used in this article, reads

$$ $\widetilde{E}(\boldsymbol{u})=\sum_{i}\int\limits_{0}^{g_{i}(u_{i})}d\tau_{i}g _{i}^{-1}(\tau_{i})-\boldsymbol{b}^{\top}\boldsymbol{g}(\boldsymbol{u})-\frac{ 1}{2}\left(\boldsymbol{g}(\boldsymbol{u})\right)^{\top}\boldsymbol{W} \boldsymbol{g}(\boldsymbol{u}),$ (5) $$

and dynamical system is precisely the same as (1). Activation functions $g_{i}$ are invertible and monotone by assumption. The first term of (5) is seemingly distinct from any term of (4). To show that this is an illusion we simplify integrals as follows

$$ $\int\limits_{0}^{g(u)}d\tau g^{-1}(\tau)=\int\limits_{g^{-1}(0)}^{u}d\rho g^{{ }^{\prime}}(\rho)\rho=-\int\limits_{g^{-1}(0)}^{u}dg(\rho)+\left.\rho g(\rho) \right|_{g^{-1}(0)}^{u}=-\int\limits_{g^{-1}(0)}^{u}dg(\rho)+ug(u),$ $$

where in the first step we used $\tau=g(\rho)$. Using this simplification and additional definition of Lagrange function $L(\boldsymbol{u}):\boldsymbol{g}(\boldsymbol{u})=\frac{\partial L(\boldsymbol{
u})}{\partial\boldsymbol{u}}$ we immediately recognize that energy function from [] is precisely the same as from [] but without the assumption that $\boldsymbol{g}$ should be invertible. New energy from [] does not formally contain the inverse of the activation function, but as we see still implies that $\boldsymbol{g}$ is invertible.

Besides insensitivity to values of dead neurons, there is another problematic fact. Proposition 1 implies that, when at least one neuron is dead, the energy function has invariant transformation $E(\boldsymbol{y})=E(\boldsymbol{y}+\boldsymbol{V}\boldsymbol{c})$. The steady state of dynamical system (1) does not have this symmetry, so, when one transforms $\boldsymbol{y}\rightarrow\boldsymbol{y}+\boldsymbol{V}\boldsymbol{c}$ and preserve energy, this corresponds to a new dynamical system with $\boldsymbol{b}\rightarrow\boldsymbol{b}-\boldsymbol{V}\boldsymbol{c}$. In other words, in the regions when dead neurons are present, energy is not sensitive to the changes in the bias term.

We summarise arguments made in this section in the proposition below

###### Proposition 2 .

For energy function (4) the following is true:

- 1.
Energy function does not depend on variables from the kernel of $\boldsymbol{g}$, so one can always assume that activation function $\boldsymbol{g}$ is invertible.
- 2.
Suppose at a given point $\boldsymbol{y}$ dead neurons are described by matrix $\boldsymbol{V}\in\mathbb{R}^{N\times k}$. In this case energy function $E(\boldsymbol{y},\widetilde{\boldsymbol{b}})$ is Lyapunov function for dynamical systems (1) with $\boldsymbol{b}=\widetilde{\boldsymbol{b}}-\boldsymbol{V}\boldsymbol{c}$ for any $\boldsymbol{c}\in\mathbb{R}^{k}_{+}$.

###### Proposition 2 .

### 3.2 Stability of dynamical system

One of the main applications of the Lyapunov function is to perform stability analysis []. This part is relevant for predictive coding and related approaches [], [] to ensure the existence of at least some steady states. Besides, when associative memory is considered, one can only recover stable steady states, so stability directly influences memory capacity. Lyapunov function also characterizes a basin of attraction for a given steady state. In the context of associative memory, basins of attraction define a measure of similarity between inputs, since inputs from the same basin result in the recovery of the same memory. Below, we will show that in the regions with dead neurons, Lyapunov function (4) alone fails to help in stability analysis.

Stability analysis with the Lyapunov function boils down to the study of the energy landscape near the steady state. Using the Taylor series, we find

$$ $E(\boldsymbol{y}^{\star}+\boldsymbol{\delta})-E(\boldsymbol{y}^{\star})= \boldsymbol{\delta}^{\top}\left.\frac{\partial E(\boldsymbol{u})}{\partial \boldsymbol{u}}\right|_{\boldsymbol{u}=\boldsymbol{y}^{\star}}+\boldsymbol{ \delta}^{\top}\left.\frac{\partial^{2}E(\boldsymbol{u})}{\partial\boldsymbol{u }^{2}}\right|_{\boldsymbol{u}=\boldsymbol{y}^{\star}+t\boldsymbol{\delta}} \boldsymbol{\delta},\,t(\boldsymbol{\delta},\boldsymbol{y}^{\star})\in(0,1),$ $$

where $\boldsymbol{y}^{\star}:\boldsymbol{W}\boldsymbol{g}(\boldsymbol{y}^{\star})-
\boldsymbol{y}^{\star}+\boldsymbol{b}=0$. Computing derivatives, we find

$$ $E(\boldsymbol{y}^{\star}+\boldsymbol{\delta})-E(\boldsymbol{y}^{\star})= \boldsymbol{\delta}^{\top}\left(\boldsymbol{\Lambda}(\boldsymbol{y}^{\star})- \boldsymbol{\Lambda}(\boldsymbol{y}^{\star})\boldsymbol{W}\boldsymbol{\Lambda} (\boldsymbol{y}^{\star})\right)\boldsymbol{\delta},\left\|\boldsymbol{\delta} \right\|_{2}\ll 1$ (6) $$

For stability in the vicinity of a steady state, it is sufficient that the matrix above is positive definite, and for instability, it is sufficient that at least one eigenvalue of this matrix is negative. In situations such as in Figure 1 when energy has a flat direction nothing can be said about the dynamics in the region with flat energy. It is easy to show that if dead neurons are present at point $\boldsymbol{u}$, matrix $\boldsymbol{V}$ lays in zero eigenspace of $\boldsymbol{\Lambda}(\boldsymbol{u})$. To show this we consider the Taylor series

$$ $\boldsymbol{g}(\boldsymbol{u}+\boldsymbol{V}\boldsymbol{c})=\boldsymbol{g}( \boldsymbol{u})+\boldsymbol{\Lambda}(\boldsymbol{u}+t\boldsymbol{V}\boldsymbol {c})\boldsymbol{V}\boldsymbol{c},\,t\in(0,1),$ (7) $$

and since $\boldsymbol{g}(\boldsymbol{u}+\boldsymbol{V}\boldsymbol{c})=\boldsymbol{g}(
\boldsymbol{u})$ we confirm that $\boldsymbol{\Lambda}\boldsymbol{V}=0$, i.e., zero eigenspace is always present(^1^11In the case when energy is flat one may hope to use LaSalle’s invariance principle [, Theorem 3.3]. Unfortunately, it is also not applicable, because flat regions are not compact, so the existence of a positive invariant set can not be ensured.)

Since energy is insufficient to analyze stability, we need to use a dynamical system (1) directly. We suppose that in the region of interest $\boldsymbol{V}$, matrix that describes dead neurons, does not change and use $\boldsymbol{y}_{d}$, $\boldsymbol{y}_{d}$ defined in the previous section. With that, one can show that dynamical system becomes

$$ $\dot{\boldsymbol{y}}_{a}(t)=\boldsymbol{P}_{a}\boldsymbol{W}\boldsymbol{g}( \boldsymbol{y}_{a}(t))-\boldsymbol{y}_{a}+\boldsymbol{P}_{a}\boldsymbol{b},\, \dot{\boldsymbol{y}}_{d}(t)=\boldsymbol{P}_{d}\boldsymbol{W}\boldsymbol{g}( \boldsymbol{y}_{a}(t))-\boldsymbol{y}_{d}+\boldsymbol{P}_{d}\boldsymbol{b},$ (8) $$

where $\boldsymbol{P}_{d}=\boldsymbol{V}\boldsymbol{V}^{\top}$ and $\boldsymbol{P}_{a}=\boldsymbol{I}-\boldsymbol{P}_{d}$. One can immediately see that dead neurons do not influence stability and, when steady state $\boldsymbol{y}_{a}^{\star}$ is reached, one can find $\boldsymbol{y}_{d}^{\star}=\boldsymbol{P}_{d}\boldsymbol{W}\boldsymbol{g}(
\boldsymbol{y}_{a}^{\star})+\boldsymbol{P}_{d}\boldsymbol{b}$. Moreover, when $\boldsymbol{y}_{a}(t)$ reaches steady state $\boldsymbol{y}_{d}(t)$ converges exponentially from the arbitrary starting point, i.e., the whole flat region is a basin of attraction.

The sufficient condition for stability/instability of $\boldsymbol{y}_{a}$ can be obtained from (8) with linearisation.
More specifically, stability is defined by matrix $\boldsymbol{S}=\boldsymbol{V}_{\perp}^{\top}\boldsymbol{W}\boldsymbol{V}_{
\perp}\boldsymbol{V}_{\perp}^{\top}\boldsymbol{\Lambda}\boldsymbol{V}_{\perp}-
\boldsymbol{I}\equiv\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{\perp}-
\boldsymbol{I}\in\mathbb{R}^{(N-k)\times(N-k)}$, where $\boldsymbol{V}_{\perp}\in\mathbb{R}^{N\times N-k}$ is a matrix with orthogonal columns such that $\boldsymbol{P}_{a}=\boldsymbol{V}_{\perp}\boldsymbol{V}_{\perp}^{\top}$. When matrix $\boldsymbol{S}$ does not have eigenvalues with a positive real part, the steady state is stable; in case there is at least one eigenvalue with a positive real part, the steady state is unstable (see [, Theorem 3.19]).

The structure of matrix $\boldsymbol{S}$ suggests that, in fact, dynamic is stable when energy function indicates that. Indeed, $\boldsymbol{\Lambda}_{\perp}^{1/2}\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{
\perp}^{1/2}$ has the same spectrum as $\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{\perp}$, so sufficient condition for $\boldsymbol{S}$ is equivalent to sufficient condition for $\boldsymbol{\Lambda}_{\perp}^{1/2}\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{
\perp}^{1/2}-\boldsymbol{I}$. Finally since $\boldsymbol{\Lambda}_{\perp}$ is full rank we find that if $\boldsymbol{\Lambda}_{\perp}\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{\perp}
-\boldsymbol{\Lambda}_{\perp}$ is negatively stable (has non-positive eigenvalues), the dynamics of $\boldsymbol{y}_{a}$ is stable. Since this matrix is a restriction of $\boldsymbol{\Lambda}\boldsymbol{W}\boldsymbol{\Lambda}-\boldsymbol{\Lambda}$ on the range of Hessian we conclude that one can analyze the stability of steady-state using energy if projector on the range of Hessian is known. It is important to note that matrix $\boldsymbol{\Lambda}\boldsymbol{W}\boldsymbol{\Lambda}-\boldsymbol{\Lambda}$ can have other flat directions not corresponding to nullspace of Hessian, these directions will cause instability, so one need to distinguish them from directions corresponding to dead neurons.

Clearly, zero modes of $\boldsymbol{\Lambda}$ do not negatively affect the spectrum of matrix $\boldsymbol{S}$. On the contrary, one may expect a stabilizing effect since Cauchy interlacing theorem [, Theorem 4.3.28, Corollary 4.3.37] ensures that the spectrum of $\boldsymbol{W}_{\perp}$ is in-between minimal and maximal eigenvalues of $\boldsymbol{W}$. So when $\lambda\left(\boldsymbol{W}\boldsymbol{\Lambda}\right)\geq 1$ it is likely that $\lambda\left(\boldsymbol{W}_{\perp}\boldsymbol{\Lambda}_{\perp}\right)<1$. To demonstrate the effect, one needs to make some assumptions about the spectrum of $\boldsymbol{W}$. Given a standard deep learning practice to use random matrices for initialization of model weights [], [], it is natural to assume that $\boldsymbol{W}$ is drawn from Gaussian orthogonal ensemble. Under this condition, one may show a stabilization effect, as demonstrated next.

###### Proposition 3 .

Suppose $\boldsymbol{W}\in\mathbb{R}^{N\times N}$ is a matrix from a Gaussian orthogonal ensemble. If $k$ neurons are dead $\mathbb{E}\left\|\boldsymbol{W}_{\perp}\right\|_{2}=\sqrt{1-\frac{k}{N}}\,
\mathbb{E}\left\|\boldsymbol{W}\right\|_{2}$ holds for large $N$.

###### Proof.

Matrix $\boldsymbol{W}_{\perp}$ can be written as $\boldsymbol{D}^{\top}\boldsymbol{O}^{\top}\boldsymbol{W}\boldsymbol{O}
\boldsymbol{D}$ where $\boldsymbol{O}$ is orthogonal full rank matrix and $\boldsymbol{D}$ is first $N-k$ columns of $\boldsymbol{I}$. Since GOE is invariant under orthogonal transformations $\boldsymbol{O}^{\top}\boldsymbol{W}\boldsymbol{O}$ is also a matrix from GOE. The effect of $\boldsymbol{D}$ is to select a principal submatrix with $N-k$ rows and columns. Given that $\boldsymbol{W}_{\perp}$ is also from GOE but with $N-k$ rows and columns. For GOE with large $N$, the distribution of leading eigenvalue quickly converges to Tracy–Widom distribution (see [] equation (50) and numerical results), so, on the average, spectral radius of a matrix with $N-k$ rows and columns is $\sqrt{2(N-k)}$ and $\mathbb{E}\left\|\boldsymbol{W}_{\perp}\right\|_{2}\big{/}\mathbb{E}\left\|
\boldsymbol{W}\right\|_{2}=\sqrt{1-\frac{k}{N}}$.
∎

We gather the main results of this section in the proposition that follows.

###### Proposition 4 .

For energy function (4) the following is true:

- 1.
Flat directions of dead neurons characterised by $\boldsymbol{V}$ form nullspace of Hessian $\boldsymbol{\Lambda}$.
- 2.
Steady state is stable if energy function predicts local stability in the range of Hessian, i.e. if matrix $\boldsymbol{\Lambda}\boldsymbol{W}\boldsymbol{\Lambda}-\boldsymbol{\Lambda}$ restricted on the range of Hessian is positive definite.
- 3.
If one performed direct optimization of the energy function and found that $\widetilde{\boldsymbol{y}}$ corresponds to the minimum, it is possible to recover the steady state of dynamical system (1) as follows: (i) compute the projector $\boldsymbol{P}_{d}$ on the nullspace of $\boldsymbol{\Lambda}$; (ii) steady state of dynamical system reads $\boldsymbol{y}=\widetilde{\boldsymbol{y}}+\boldsymbol{P}_{d}\left(\boldsymbol{
W}\boldsymbol{g}(\widetilde{\boldsymbol{y}})+\boldsymbol{b}\right)$.
- 4.
When steady state $\left(\boldsymbol{y}_{a}^{\star},\boldsymbol{y}_{d}^{\star}\right)$ is stable, the whole non-compact flat region $\left(\boldsymbol{y}_{a}^{\star},\boldsymbol{y}_{d}^{\star}+\boldsymbol{V}
\boldsymbol{c}\right)$ is basin of attraction.

###### Proposition 3 .

###### Proof.

###### Proposition 4 .

## 4 Associative memories without dead neurons

We have seen that dead neurons present certain problems for energy function (4): (i) flat regions make optimization and analysis of energy cumbersome; (ii) one needs to distinguish between harmless flat direction of Hessian and other flat direction of energy that can compromise stability; (iii) energy do not contain full information and one need to build a projector on the nullspace of Hessian to restore the whole state. The last point implies that one needs Hessian even when stability analysis is not performed and one merely wants to compute a local minimum of energy which can often be done successfully with first-order methods.

To avoid these difficulties we will slightly modify the dynamics of associative memory (1) to minimally alter steady states and get a better energy function. The resulting system that we propose reads:

$$ $\dot{\boldsymbol{u}}(t)=\boldsymbol{R}(\boldsymbol{u})\left(\boldsymbol{g}( \boldsymbol{W}\boldsymbol{u}(t)+\boldsymbol{b})-\boldsymbol{u}(t)\right),\, \boldsymbol{u}(0)=\boldsymbol{u}_{0},$ (9) $$

where $\boldsymbol{R}(\boldsymbol{u})$ is a matrix-valued function to be specified later.

If one assumes that it is possible to select $\boldsymbol{R}(\boldsymbol{u})\neq 0$ for all $\boldsymbol{u}$, the condition that we will verify later, steady states of the newly introduced system (9) follows from the old one with affine transformation $\boldsymbol{y}=\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}$. This means all architectures possible with the old model (1) are also possible with the new one (9).

Dynamical systems (9) and (1) have equivalent steady states, but (9) allows to construct of a large set of Lyapunov functions with good properties in a systematic manner.

###### Proposition 5 .

For a parametric energy function

$$ $\begin{split}&E(\boldsymbol{u};\alpha,\beta,\gamma,\boldsymbol{S})=\alpha E_{1 }(\boldsymbol{u})+\beta E_{2}(\boldsymbol{u})+\gamma E_{3}(\boldsymbol{u}; \boldsymbol{S}),\alpha,\beta,\gamma\in\mathbb{R},\\ &E_{1}(\boldsymbol{u})=\left(\boldsymbol{u}^{\top}\boldsymbol{W}\boldsymbol{g} \left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)-L\left(\boldsymbol{W} \boldsymbol{u}+\boldsymbol{b}\right)-\frac{1}{2}\boldsymbol{g}\left( \boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)^{\top}\boldsymbol{W} \boldsymbol{g}\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)\right), \\ &E_{2}(\boldsymbol{u})=\frac{1}{2}\boldsymbol{u}^{\top}\boldsymbol{W} \boldsymbol{u}-L\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right),\,E_{ 3}\left(\boldsymbol{u};\boldsymbol{S}\right)=\frac{1}{2}\left(\boldsymbol{u}- \boldsymbol{g}(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b})\right)^{\top} \boldsymbol{S}\left(\boldsymbol{u}-\boldsymbol{g}(\boldsymbol{W}\boldsymbol{u} +\boldsymbol{b})\right)\end{split}$ (10) $$

the following is true:

- 1.
$E(\boldsymbol{u};\alpha,\beta,\gamma,\boldsymbol{S})$ is a Lyapunov function for dynamical system (9) if one can find $\boldsymbol{Q}\geq 0$, $\boldsymbol{R}(\boldsymbol{u})$, $\boldsymbol{S}=\boldsymbol{S}^{\top}$ such that
$\boldsymbol{R}^{\top}\boldsymbol{F}+\boldsymbol{F}^{\top}\boldsymbol{R}\geq
\boldsymbol{Q},\boldsymbol{F}(\boldsymbol{u})=\left(\boldsymbol{W}\boldsymbol{
\Lambda}\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)\left(\alpha
\boldsymbol{W}-\gamma\boldsymbol{S}\right)+\beta\boldsymbol{W}+\gamma
\boldsymbol{S}\right)$
(11)
holds for all $\boldsymbol{u}$.
- 2.
For arbitrary parameters $\alpha,\beta,\gamma,\boldsymbol{S}$ one can always find orthogonal matrix $\boldsymbol{R}(\boldsymbol{u})$ such that $E(\boldsymbol{u};\alpha,\beta,\gamma,\boldsymbol{S})$ is a Lyapunov function for dynamical system (9).
- 3.
$E_{2}(\boldsymbol{u})$ and $E_{3}(\boldsymbol{u};\boldsymbol{S})$ does not have flat directions corresponding to dead neurons.
- 4.
Sufficient condition for stability of steady-state $\boldsymbol{u}^{\star}:\boldsymbol{u}^{\star}=\boldsymbol{g}\left(\boldsymbol{
W}\boldsymbol{u}^{\star}+\boldsymbol{b}\right)$ is
$\boldsymbol{F}(\boldsymbol{u}^{\star})\left(\boldsymbol{I}-\boldsymbol{\Lambda
}\left(\boldsymbol{W}\boldsymbol{u}^{\star}+\boldsymbol{b}\right)\boldsymbol{W
}\right)+\left(\boldsymbol{I}-\boldsymbol{W}\boldsymbol{\Lambda}\left(
\boldsymbol{W}\boldsymbol{u}^{\star}+\boldsymbol{b}\right)\right)\left(
\boldsymbol{F}(\boldsymbol{u}^{\star})\right)^{\top}>0$
(12)

###### Proof.

Observe that $\frac{\partial L\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)}{
\partial\boldsymbol{u}}=\boldsymbol{W}\boldsymbol{g}\left(\boldsymbol{W}
\boldsymbol{u}+\boldsymbol{b}\right)$ and $\frac{\partial\boldsymbol{g}\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}
\right)}{\partial\boldsymbol{u}}=\boldsymbol{\Lambda}\left(\boldsymbol{W}
\boldsymbol{u}+\boldsymbol{b}\right)\boldsymbol{W}$.

- 1.
Using the identities above we find
$\begin{split}&\frac{\partial E_{1}}{\partial\boldsymbol{u}}=\boldsymbol{W}
\boldsymbol{\Lambda}\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)
\boldsymbol{W}\left(\boldsymbol{u}-\boldsymbol{g}(\boldsymbol{W}\boldsymbol{u}
-\boldsymbol{b})\right),\,\frac{\partial E_{2}}{\partial\boldsymbol{u}}=
\boldsymbol{W}\left(\boldsymbol{u}-\boldsymbol{g}(\boldsymbol{W}\boldsymbol{u}
-\boldsymbol{b})\right),\\
&\frac{\partial E_{3}}{\partial\boldsymbol{u}}=\left(\boldsymbol{I}-
\boldsymbol{W}\boldsymbol{\Lambda}\left(\boldsymbol{W}\boldsymbol{u}+
\boldsymbol{b}\right)\right)\boldsymbol{S}\left(\boldsymbol{u}-\boldsymbol{g}(
\boldsymbol{W}\boldsymbol{u}-\boldsymbol{b})\right).\\
\end{split}$
Multiplying gradients on $\alpha,\beta,\gamma$ and adding them gives $\boldsymbol{F}(\boldsymbol{u})$. Since $\dot{\boldsymbol{u}}=\boldsymbol{R}(\boldsymbol{u})\boldsymbol{f}(\boldsymbol{
u}(t))$ where $\boldsymbol{f}(\boldsymbol{u}(t))=\boldsymbol{u}-\boldsymbol{g}(\boldsymbol{W}
\boldsymbol{u}-\boldsymbol{b})$ from $\dot{E}=\dot{\boldsymbol{u}}^{\top}\frac{\partial E}{\partial\boldsymbol{u}}$ we obtain
$\dot{E}=-\boldsymbol{f}(\boldsymbol{u})^{\top}\left(\boldsymbol{R}^{\top}
\boldsymbol{F}(\boldsymbol{u})+\left(\boldsymbol{F}(\boldsymbol{u})\right)^{
\top}\boldsymbol{R}\right)\boldsymbol{f}(\boldsymbol{u}).$
If one can find positive semidefinite matrix $\boldsymbol{Q}$ such that $\boldsymbol{R}^{\top}\boldsymbol{F}(\boldsymbol{u})+\left(\boldsymbol{F}(
\boldsymbol{u})\right)^{\top}\boldsymbol{R}\geq\boldsymbol{Q}$, energy is non-increasing on trajectories since in this case $\dot{E}\leq-\boldsymbol{f}^{\top}\boldsymbol{Q}\boldsymbol{f}\leq 0$
- 2.
Consider polar decomposition of matrix $\boldsymbol{F}(\boldsymbol{u})=\boldsymbol{O}(\boldsymbol{u})\boldsymbol{P}(
\boldsymbol{u})$ and take $\boldsymbol{R}(\boldsymbol{u})=\boldsymbol{O}(\boldsymbol{u})$. Since $\boldsymbol{P}(\boldsymbol{u})\geq 0$ we have $\boldsymbol{R}^{\top}\boldsymbol{F}(\boldsymbol{u})+\left(\boldsymbol{F}(
\boldsymbol{u})\right)^{\top}\boldsymbol{R}=2\boldsymbol{P}(\boldsymbol{u})\geq
0$ and $\dot{E}\leq 0$.
- 3.
From the proof of Proposition 1 we know that if $\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\rightarrow\boldsymbol{W}
\boldsymbol{u}+\boldsymbol{b}+\boldsymbol{V}\boldsymbol{c}$, Lagrange function shifts on $\boldsymbol{g}\left(\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\right)
\boldsymbol{V}\boldsymbol{c}$. The first term in the energy function (10) is quadratic and does not contain $\boldsymbol{g}$ so in general energy $E_{2}$ does not have flat directions. Similarly, $E_{3}$ has a form $\boldsymbol{f}(\boldsymbol{u})^{\top}\boldsymbol{S}\boldsymbol{f}(\boldsymbol{
u})$ where $\boldsymbol{f}$ is not invariant under transformation $\boldsymbol{W}\boldsymbol{u}+\boldsymbol{b}\rightarrow\boldsymbol{W}
\boldsymbol{u}+\boldsymbol{b}+\boldsymbol{V}\boldsymbol{c}$ meaning $E_{3}$ is not invariant as well.
- 4.
Since $\frac{\partial E}{\partial\boldsymbol{u}}=\boldsymbol{F}(\boldsymbol{u})
\boldsymbol{f}(\boldsymbol{u})$ and $\boldsymbol{f}(\boldsymbol{u}^{\star})=0$ we find for sufficiently small $\boldsymbol{\delta}$
$E(\boldsymbol{u}^{\star}+\boldsymbol{\delta})-E(\boldsymbol{u}^{\star})=
\boldsymbol{\delta}^{\top}\boldsymbol{F}(\boldsymbol{u}^{\star})\left.\frac{
\partial\boldsymbol{f}(\boldsymbol{u})}{\partial\boldsymbol{u}}\right|_{
\boldsymbol{u}=\boldsymbol{u}^{\star}}\boldsymbol{\delta}.$
For stability one needs this quadratic form to be positive definite, which gives (12).

∎

Proposition 5 allows one to make several useful conclusions.

First, the construction of the Lyapunov function boils down to the analysis of matrix inequality (11). In general, this inequality is nonlinear, but in particular cases, it reduces to a Sylvester equation which can be systematically analyzed []. Below we will provide several explicit choices of $\boldsymbol{R}$ that result in a valid Lyapunov function.

The second point of Proposition 5 shows that suitable $\boldsymbol{R}$ always exist. Moreover, this $\boldsymbol{R}$ does not alter the steady state, since it is an orthogonal matrix. Whether this choice of $\boldsymbol{R}$ is practical depends on the parametrization of weights and other details of the model.

The third point shows that unless $\beta=\gamma=0$, the energy function does not have a flat direction, and the fourth point gives a sufficient condition for the stability of a particular state. This part is harder to analyze in general, so we will discuss this condition for selected examples below.

Finally, Proposition 5 does not require Hessian to be positive definite which allows one to use a richer set of activation functions.

###### Example 5 ( $\alpha=\gamma=0$ , $\beta=1$ ) .

In this case $E=E_{1}$ condition (11) becomes $\boldsymbol{R}^{\top}\boldsymbol{W}+\boldsymbol{W}\boldsymbol{R}\geq 0$. There are many strategies to select $\boldsymbol{R}$ and $\boldsymbol{W}$:

- 1.
The simplest choice is to take $\boldsymbol{R}=\boldsymbol{W}$ since in this case condition (11) reduces to $\boldsymbol{W}^{2}\geq 0$ which is true for arbitrary symmetric matrix.
- 2.
Unfortunately, if matrix $\boldsymbol{W}$ has low rank and one takes $\boldsymbol{R}=\boldsymbol{W}$, dynamical system (9) can lead to false memories whenever $\boldsymbol{f}(\boldsymbol{u}(t))$ reaches a nullspace of $\boldsymbol{W}$. To exclude this possibility one can parametrize $\boldsymbol{W}=\boldsymbol{O}\boldsymbol{P}$ in terms of its polar decomposition and select $\boldsymbol{R}=\boldsymbol{O}$. With that choice condition (11) becomes $\boldsymbol{P}\geq 0$ which automatically follows from polar decomposition.
- 3.
We can restrict $\boldsymbol{W}$ to be positive definite by taking $\boldsymbol{W}=\boldsymbol{K}\boldsymbol{K}^{\top}$ for some full-rank $\boldsymbol{K}$. In this case, a large set of positive definite matrices $\boldsymbol{R}$ exists such that $\boldsymbol{R}\boldsymbol{W}+\boldsymbol{W}\boldsymbol{R}\geq 0$. More specifically, one needs to restrict the condition number of $\boldsymbol{R}$ as explained in [].
- 4.
If $\boldsymbol{W}$ is positive definite the other option is to select arbitrary $\boldsymbol{Q}>0$ and consider Lyapunov equation $\boldsymbol{R}\boldsymbol{W}+\boldsymbol{W}\boldsymbol{R}=\boldsymbol{Q}$ which is known to have unique solution for arbitrary $\boldsymbol{Q}>0$. One can explicitly provide it in many forms [], [], for example $\boldsymbol{R}=\int_{0}^{\infty}d\tau\exp(-\tau\boldsymbol{W})\boldsymbol{Q}
\exp(-\tau\boldsymbol{W})$.

Condition for the stability of state (12) in this case reduces to $\boldsymbol{W}-\boldsymbol{W}\boldsymbol{\Lambda}\left(\boldsymbol{W}
\boldsymbol{u}^{\star}+\boldsymbol{b}\right)\boldsymbol{W}>0$. Note that nullspace of $\boldsymbol{\Lambda}$ does not compromise stability since it does not transfer to nullspace of $\boldsymbol{W}-\boldsymbol{W}\boldsymbol{\Lambda}\boldsymbol{W}$.

Interestingly, one can also construct associative memory with non-symmetric weights as explained in the next two examples.

###### Example 6 ( $\alpha=\beta=0$ , $\gamma=1$ and $\boldsymbol{W}\neq\boldsymbol{W}^{\top}$ ) .

$E_{3}$ is the only energy function that is defined for $\boldsymbol{W}\neq\boldsymbol{W}^{\top}$. If we account for that in Proposition 5, existence condition (11) becomes $\boldsymbol{R}^{\top}\left(\boldsymbol{I}-\boldsymbol{W}^{\top}\boldsymbol{
\Lambda}\right)\boldsymbol{S}+\boldsymbol{S}\left(\boldsymbol{I}-\boldsymbol{
\Lambda}\boldsymbol{W}\right)\boldsymbol{R}\geq 0$. Two examples of suitable $\boldsymbol{R}$, $\boldsymbol{S}$, $\boldsymbol{W}$ are given below:

- 1.
The simplest choice is to take $\boldsymbol{S}=\boldsymbol{I}$ and $\boldsymbol{R}=\boldsymbol{I}-\boldsymbol{W}^{\top}\boldsymbol{\Lambda}$. One also may ensure that $\boldsymbol{R}$ is invertible with $\left\|\boldsymbol{W}\right\|_{2}<\left\|\boldsymbol{\Lambda}\right\|_{2}^{-1}$.
- 2.
Another possible choice is to take $\boldsymbol{R}=\boldsymbol{\Lambda}$, and ensure $\boldsymbol{\Lambda}-\frac{1}{2}\boldsymbol{\Lambda}\left(\boldsymbol{W}+
\boldsymbol{W}^{\top}\right)\boldsymbol{\Lambda}\geq 0$, which can be simplified to $\omega(\boldsymbol{W})\leq\lambda_{\min}(\boldsymbol{\Lambda})\big{/}\lambda_{
\max}(\boldsymbol{\Lambda}^{2})$ where $\omega(\boldsymbol{W})$ is numerical abscissa $\omega(\boldsymbol{W})=\lambda_{\max}(\boldsymbol{W}+\boldsymbol{W}^{\top})
\big{/}2$. For example, if $\boldsymbol{\Lambda}\geq 0$ any $\boldsymbol{W}$ with non-positive real part of the spectrum suffices.

###### Example 7 (associative memory with smooth Leaky ReLU) .

Here we build a concrete example of hierarchical associative memory [] with smooth Leaky ReLU [] to avoid dealing with ODE having a non-smooth right-hand side. The activation function reads $\boldsymbol{g}(\boldsymbol{u})=\boldsymbol{u}\left(5+3\text{erf}\left(3
\boldsymbol{u}\big{/}8\right)\right)\big{/}8$ and its derivative is $\boldsymbol{g}^{{}^{\prime}}(\boldsymbol{u})=\boldsymbol{g}(\boldsymbol{u})
\big{/}\boldsymbol{u}+9\boldsymbol{u}\exp\left(-9\boldsymbol{u}^{2}\big{/}64
\right)\big{/}\left(32\sqrt{\pi}\right)$, where all operations are pointwise and erf is error function. Weights and state vector are selected as in Example 1 but without requirement $\boldsymbol{W}_{ii+1}=\boldsymbol{W}_{i+1i}^{\top}$. After that we select $\boldsymbol{R}=\boldsymbol{I}-\boldsymbol{W}^{\top}\boldsymbol{\Lambda}$ and obtain the following dynamical system

$$ $\dot{\boldsymbol{u}}(t)=\left(\boldsymbol{I}-\boldsymbol{W}^{\top}\odot \boldsymbol{g}^{{}^{\prime}}(\boldsymbol{W}\boldsymbol{u}(t)+\boldsymbol{b}) \right)\left(\boldsymbol{g}(\boldsymbol{W}\boldsymbol{u}(t)+\boldsymbol{b})- \boldsymbol{u}(t)\right),\,\boldsymbol{u}(0)=\boldsymbol{u}_{0}.$ $$

For this system Lyapunov function

$$ $E_{2}=\frac{1}{2}\left(\boldsymbol{g}(\boldsymbol{W}\boldsymbol{u}(t)+ \boldsymbol{b})-\boldsymbol{u}(t)\right)^{\top}\left(\boldsymbol{g}( \boldsymbol{W}\boldsymbol{u}(t)+\boldsymbol{b})-\boldsymbol{u}(t)\right)$ $$

is non-increasing on trajectories. Matrix $\left(\boldsymbol{I}-\boldsymbol{W}^{\top}\odot\boldsymbol{g}^{{}^{\prime}}(
\boldsymbol{W}\boldsymbol{u}(t)+\boldsymbol{b})\right)$ can in principle have non-trivial nullspace. To avoid this one observe that $\left\|\boldsymbol{W}\right\|_{2}\leq\max_{i}\left\|\boldsymbol{W}_{ii+1}
\right\|_{2}+\max_{i}\left\|\boldsymbol{W}_{ii-1}\right\|_{2}$ (the bound is tight) and take $\boldsymbol{W}_{ii+1}=\widetilde{\boldsymbol{W}}_{ii\pm 1}\big{/}\left(2\left
\|\widetilde{\boldsymbol{W}}_{ii\pm 1}\right\|_{2}\right)$, for some $\widetilde{\boldsymbol{W}}$ with the same block structure. Note, that in this construction $\boldsymbol{\Lambda}$ is not positive definite, and $\boldsymbol{W}$ is not symmetric.

All theoretical statements that we derived are valid only for modified dynamical system 9. However, it is easy to see that energy function $E_{3}$ from Proposition 5 can be adopted for the original temporal dynamics (1). The resulting energy function and dynamical system reads

$$ $\displaystyle\dot{\boldsymbol{y}}(t)=\boldsymbol{R}(\boldsymbol{y}(t))\left( \boldsymbol{W}\boldsymbol{g}(\boldsymbol{y}(t))-\boldsymbol{y}(t)+\boldsymbol{ b}\right),\,\boldsymbol{y}(0)=\boldsymbol{y}_{0},$ (13) $\displaystyle E_{3}(\boldsymbol{u})=\frac{1}{2}\left(\boldsymbol{W}\boldsymbol {g}(\boldsymbol{y}(t))-\boldsymbol{y}(t)+\boldsymbol{b}\right)^{\top} \boldsymbol{S}\left(\boldsymbol{W}\boldsymbol{g}(\boldsymbol{y}(t))- \boldsymbol{y}(t)+\boldsymbol{b}\right).$ (14) $$

We used temporal dynamics (13) with $\boldsymbol{R}=\boldsymbol{I}-\boldsymbol{W}\boldsymbol{\Lambda}(\boldsymbol{u})$ and energy function (14) with $\boldsymbol{S}=\boldsymbol{I}$ to generate results given in Figure 1. One can also prove results similar to Proposition 5 for energy function (14) but since the extension is straightforward we do not pursue this further.

###### Proposition 5 .

###### Proof.

###### Example 5 ( α = γ = 0 𝛼 𝛾 0 \alpha=\gamma=0 italic_α = italic_γ = 0 , β = 1 𝛽 1 \beta=1 italic_β = 1 ) .

###### Example 6 ( α = β = 0 𝛼 𝛽 0 \alpha=\beta=0 italic_α = italic_β = 0 , γ = 1 𝛾 1 \gamma=1 italic_γ = 1 and 𝑾 ≠ 𝑾 ⊤ 𝑾 superscript 𝑾 top \boldsymbol{W}\neq\boldsymbol{W}^{\top} bold_italic_W ≠ bold_italic_W start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) .

###### Example 7 (associative memory with smooth Leaky ReLU) .

## 5 Conclusions

We describe the effect of dead neurons on the energy landscape of associative memory, analyze the consequences for stability, and provide several remedies, including new dynamical systems with good energy functions. We think it is appropriate to discuss the overall significance of the Lyapunov function for associative memory. Is it necessary to have this function at all? How is this function used in practice?

One may observe that, currently, the Lyapunov function is underutilized. As a rule, one uses several steps or even a single step of temporal dynamics of ODE during training and inference [], [], [], [], []. The role of Lyapunov’s function is merely to provide comfort that some steady states may exist somewhere. As we show in this article, it is often a false comfort.

It is important to note, that the Lyapunov function in itself does not ensure stability: (i) it may be the case that no steady state exists, (ii) limit cycles may still be present, (iii) all steady states may be unstable. Besides that, for any steady state, the Lyapunov function can be constructed (by solving the Lyapunov equation) even when the “global” Lyapunov function is unavailable.

The Lyapunov function, as it is currently considered, is not of huge help. We can suggest several appropriate use cases: (i) model parameters may be adjusted on the training stage to make the Lyapunov function unstable for particular states – something that can not be easily done by integrating the dynamical system alone. This may help to learn from negative and adversarial examples [], []; (ii) Lyapunov function may help in theoretical understanding of memory capacity, e.g., in the present article we have already shown that a whole flat direction corresponds to the basin of attraction, and can not support more than a single memory; (iii) Lapynov function may be directly used to compute and manipulate basins of attraction, potentially speeding up learning and making memory more robust to adversarial attacks.

All in all, we think that the Lyapunov function is a powerful tool that can lead to novel theoretical results and practical learning techniques in the field of associative memory. We hope that our results will inspire further research in this direction.

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: 1. I am not sure I understood the derivation in line 284. Why does the Taylor series terminate after the second term? 
2. It seems to me that the model presented in equation 9 can be obtained from the Krotov-Hopfield model by introducing a new variable $\mathbf{y}=\mathbf{Wu}+\mathbf{b}$. Are these two classes of models equivalent under this change of variables, or does the introduction of matrix R make equation 9 more general than Krotov and Hopfield?
3. In all the derivations $\mathbf{b}$ is assumed to be time-independent. This is fine (same is true in Krotov-Hopfield). I am wondering though, if anything could be said about situations when $\mathbf{b}$ depends on time? E.g., could energy function (10) still guarantee convergences for some properties of $\mathbf{b(t)}$?
soundness: 3
strengths: Dense Associative Memory models have several possible formulations and admit studies of a broad class of AI models used in practice. At the same time, their Lyapunov (energy) functions remain underutilized and the shape of those energy landscapes remains under investigated. This paper tackles this exact question and analyses the stability of the dynamics and the sensitivity of the energy with respect to the state of the neurons from a novel angle. The problem of dead neurons can be a serious obstacle for training these models at scale in interesting AI applications. This work formalizes this problem, and offers a possible solution by identifying non-dead subspaces and proposing a novel class of energy functions. The paper is generally well-written, although there are small issues, which I explain below. The results presented in this work are novel and relevant to the ICLR community.
summary: The work investigates Lyapunov functions and stability in the Krotov and Hopfield models of associative memory. It is found that the conventional formulation suffers from a problem of dead neurons, which arises when some of the neurons’ activation functions saturate. In this situation the energy becomes independent of the state of those neurons. The authors propose a projection of the original class of models on a subspace of non-dead neurons, analyze the stability of those models, and propose a novel class of Lyapunov functions that describe those models.
weaknesses: There are two issues with the way how authors introduce the problem of dead neurons (example 2, 3, 4). First, in the Krotov-Hopfield formalism, one needs to check two things: 
1. Positive definiteness of the Hessian of the Lagrangian. 
2. Boundness of the energy function from below. 

Only when both conditions are satisfied the models under consideration have stable fixed point dynamics. The authors rightfully focus on the first requirement in their presentation, but somehow the second one is not discussed enough. For instance, please notice that the model studied in Example 2 does not have an energy bounded from below. This means that one can choose a matrix $\mathbf{W}$ such that the energy will keep decreasing to minus infinity with time, and the model will never reach a fixed point. The lower-boundness of the energy needs to be checked and discussed in the paper. 


Second, Example 4 represents a situation when Lagrangian has a symmetry. Please notice (this is acknowledged in lines 194-195) that in this case there are no dead neurons, all the neurons are alive, it’s just the energy function is insensitive to the states of the variables $\mathbf{y}$ along the constant shifts of all pre-activations. I am wondering to what extent it makes sense to dub this specific example as a “dead-neuron” problem. The authors rightfully acknowledge this in lines 192-198, but I would invite the authors to consider restructuring the presentation of this example and give this specific phenomenon (Example 4) a different name, e.g. “symmetry”, “flat directions”, etc., not the dead-neuron problem. 

Some typos: 
1. Line 507: $E_3(\mathbf{u})$ -> $E_3(\mathbf{y})$
2. Line 509: $\Lambda(\mathbf{u})$ -> $\Lambda(\mathbf{y})$
3. Line 156-157 - subscript in $W_{21}$. 
4. Line 535: Lapynov-> Lyapunov

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 2
questions: Are there specific tasks in which you could measure the advantage of the newly proposed energy function? For example, testing how many stored memories can you perfectly retrieve, or the capacity of the model? Similarly to the many provided in Millidge et al.?
soundness: 3
strengths: The work is well structured: it identifies a gap in the literature, and theoretically addresses it. The gap is also a significant one, as flat regions of steady states can negatively affect the retrieval of specific memories.

The stability analysis is interesting, and it seems well made. Similarly, I share the belief of the authors about the underutilisation of the Lyapunov function in the literature. Actually, I would move the discussion in the conclusion (“*The role of Lyapunov’s function is merely to provide comfort that some steady states may exist somewhere. As we show in this article, it is often a false comfort.*”) in the introduction, to make the claim of the paper clearer. 

Despite not being sure I was able to follow the math all the way, I believe the solution proposed by the authors to be correct.
summary: This work expands on the literature of modern Hopfield networks, and that of Dense associative memories. In detail, it identifies a drawbacks of the current approach developed by Krotov and Hopfield, that is, the presence of ‘dead neurons’ that cause regions of constant energy. The authors then address this problem by defining a new family of Lyapunov functions that is a variation of the original one, but does not suffer from the problem of dead neurons causing flat regions.
weaknesses: The biggest cons of this work is the clarity. I found it hard to follow the computations, theorems, claims, and their consequences. The 10 pages are very dense. I would: 

1) Add figures that make some concepts clearer; 

2) Move some non-interesting proofs to the appendix, and leave in the main body only the computations needed to understand concepts (you could even provide proof sketches).

As I did not manage to follow the paper perfectly, I will also provide a low confidence score.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: There seems to be a typo on line 044: "The most cluical requirement" should be "The most crucial requirement".

Equations 1 through 4 --- what do these correspond to in Krotov and Hopfield 2020? Looking through the cited paper, there are no equations of a similar form, have the authors of this paper rewritten something from Krotov and Hopfield 2020 in a more convenient form? If so, perhaps a brief description of what has been changed from Krotov and Hopfield 2020 (a notation translation guide) would benefit this paper. Further, Equations 1 through 4 appear to actually consist of more than four equations, some of which are constraints (such as W = W transpose). Cleaning up the formatting here such that each equation is on one line with a separate number may benefit the paper.

At the start of Section 2 the paper makes several claims about the properties of Equations 1 through 4, including important properties of the energy function in Equation 4. However, these claims are not obvious to me. Is there a reason that the "structural assumptions" of Equation 2 (is that the first or second equation on line two?) ensures the energy function exists? This discussion could be expanded slightly for readers who are not familiar with the equations set forth here.

The next paragraph continues these discussions and makes claims about the dynamical systems described by Equation 1. However, again I can not immediately see why the steady states of these differential equations can be used as memory vectors --- could this be discussed further or given a citation?

Example 1 seems somewhat ill-defined in the edge cases. The given differential equation for y_i does not apply for y_1; what is the value of the weight matrix W_{10}? Should we treat this as 0? This would line up with my expectation of the MLP with feedback connections, but the formalization does not support it. Perhaps some discussion of the edge cases here would be beneficial.

There is a slight typesetting error on line 156: The equation is currently rendered like "$ W_{12} = W_{2}1^\top $", rather than "$ W_{12} = W_{21}^\top $" i.e. the "1" on the second matrix W is not subscript. 

Example 2 is the first time that the term "dead neuron" is linked directly to the ReLU activation. Dead neurons are referenced to Lu et al. 2019 in the introduction (line 091) but perhaps a short introduction to the term to ground this work would help a reader understand where the discussion is heading. Especially considering the term is then applied to other activations in Examples 3 and 4. What exactly is meant by a neuron being dead? This could be added to the introduction. Reading on, the definition of a dead neuron appears quite late in Section 2, which is already substantially through the paper. Is there some way to introduce the key concepts of the paper earlier?

The paper seems to have a significant focus on the energy function being non-discriminative on a non-compact region of state space. It may benefit a reader's understanding to have a short discussion on what this means, especially in the context of associative memories and physics inspired models.

The proof for Proposition 1 is somewhat difficult to follow. The Taylor expansion of the Lagrangian seems to make the implicit assumption that $ L(y) = 0.5 g(y)^2 $, which perhaps could be stated explicitly in the proof even though this is a common formulation. Past this, it is not clear to me why the second term shifts by the specified quantity, and how this is compensated by a similar shift in the first term. Some more verbose discussion here could help readers follow the proof.

Much of the body of this paper hinges on the formalization of Equations 1-4. Without having this section very clearly laid out for the reader, the remainder of the paper is often difficult to follow. Is it possible to rework this initial Section 2 to improve clarity? In particular, stating notation clearly early in the paper would be very helpful. Following sections would then be able to reference back to the key statements in Section 2 with minimal confusion.

Again for Equation 5 it is not clear what Lyapunov function you are referencing in Hopfield 1984. This looks to be most similar to Equation 7 or 11 in the 1984 paper, but the lack of notation formalization in this work makes it difficult to draw parallels.

Much of Section 3 and 4, (e.g. Section 3.2), may benefit from being written into a longer paper where the authors can be more verbose in explaining each step of the mathematics.  Currently, the paper is very terse in its discussion, which is understandable considering the page limit of the conference, but this does not help a reader understand the key aspects of this mathematically dense manuscript.

Near the start of Section 4, where we discuss the value of R(u), are we assuming that no value of the matrix R(u) is zero, or that the matrix is not equal to the zero matrix (i.e. all values zero)? The manuscript is slightly unclear on this.
soundness: 2
strengths: The work seems rigorous in its discussions, providing proofs of propositions and analyzing the proposed energy functions extensively. The work appears original, although it is mostly building off previous associative memory literature and work on dead neurons. The paper appears to be reasonably significant in the field of associative memories, and if the claims of the impact of dead neurons are true, this could represent a significant step in the analysis of neural ODEs and similar dynamical systems.
summary: This paper builds on the work of Krotov and Hopfield 2020 looking at associative memories, with particular emphasis on the classical continuous Hopfield network (Hopfield 1984), the Dense Associative Memory (Krotov and Hopfield 2016), the modern continuous Hopfield network (Ramsauer et al. 2020), and the energy transformer (Hoover et al. 2023). More broadly, this paper looks at associative memories as dynamical systems, similar to neural ODEs, and investigates the impact of dead neurons on such systems. A dead neuron in this context refers to a neuron which has no effect on the model behavior due to always taking values that are in some way constant. Dead neurons can arise from many situations, but two such classes discussed in Section 2 are saturated neurons and symmetric activation functions. This paper then analyzes how dead neurons impact the degrees of freedom of a dynamical system and the Lyapunov function of the models. In particular, it is shown that the associative memories in current literature may suffer from steady states that are not stable i.e. there are dimensions of the state space (associated with each dead neuron) that are free to vary without affecting the energy. The paper then introduces new energy functions that alleviates some of the issues relating to dead neurons and shows all architectures explained by the functions in Krotov and Hopfield 2020 are also explainable using the new functions by an affine transformation of the neuron activities.
weaknesses: The basis of the paper, Section 2, is confusing and difficult to match up to other literature on associative memories. In particular, Equations 1 through 4, which are fundamental to the remainder of the paper, are stated to be from Krotov and Hopfield 2020, but do not appear in the form given in this paper. A rewriting of Section 2 would be beneficial to improve the clarity of the work and aid reader's understanding. Rewriting of subsequent sections to reference a clarified Section 2 would also help a reader follow each point made by this work.

The notation used in the paper is unique to other papers on associative memories, which in itself is not an issue, but there is no explanation of the notation. This is a problem when discussing other literature, as matching the discussions in this paper to other referenced papers becomes nearly impossible. In many sections, steps are taken that are not obvious to the reader and may impact the ability to follow this work. There are numerous grammatical errors throughout the manuscript that can obfuscate the meaning of sentences or entire paragraphs. It is difficult to verify the claims made in this paper due to the difficult-to-follow nature of each section.

The key ideas of the paper seem sufficient for publication, but would need to be presented in a more understandable format.

Meta-review: **Summary:**
This paper studies the Krotov-Hopfield (KH) model of associative memory (Krotov-Hopfield, 2020), and points out that the energy function of their model suffers from the problem of dead neurons, implying that the energy may have flat directions, which in turn may be harmful in investigating stability of the model. This paper then proposes an alternative formulation which uses not the **pre**-activation variables but the **post**-activation variables as the state variables of the model. The proposed formulation is shown to resolve the dead-neuron problem, while maintaining other plausible properties of the KH model.

**Strengths:**
This paper studies the problem of dead neurons in the existing KH model of associative memory, and proposes how to resolve it via adopting the post-activation variables as the state variables of the model instead of the pre-activation variables in the HK model.

**Weaknesses:**
- Most reviewers raised concern about presentation quality and clarity of this paper.
- I noticed several minor errors in the manuscript. See the list below.

**Reasons:**
The reviewers evaluated this paper positively, though some of them expressed concerns about the presentation quality of this paper. Upon my own reading of this paper, I also noticed several points which would require revision: see the following list.

Minor points:
- Page 3, equations (2)-(4): The function $L(y)$ appears without proper description. The fact that $L(y)$ is some convex function of $y$, which defines the activation function $g$ via equation (2) should be explicitly stated.
- Page 3, line 156: $W_21^\top$ → $W_{21}^\top$
- Page 3, line 158: The double indices $ii-1$ and $ii+1$ are not easy to parse. I would prefer $W_{i,i-1}$ and $W_{i,i+1}$. The same applies also to page 10, lines 487 and 497 as well.
- Page 3, line 162: it allowed (the) authors
- page 4, line 165: Authors → The authors
- Page 4, Example 3: The sigmoid function saturates only in the asymptotic where its argument goes to $\pm\infty$, so that the statement in this example would be imprecise.
- Page 4, Example 4: It would be better to use the expression $\tilde{y}(c)$ throughout the example, rather than mixing the expressions $\tilde{y}(c)$ and $\tilde{y}$, in order to make the presentation coherent.
- Page 3, lines 202-204: As mentioned above, hyperbolic tangent and Gaussian saturate only asymptotically.
- Page 4, Proposition 1: $c$ appearing in the definition of the subspace $\mathcal{D}$ should be in boldface.
- Page 5, line 225; page 6, line 302: $VV^\top$ should be replaced with $V(V^\top V)^{-1}V^\top$, because it is **not** assumed that $V$ has orthonormal columns.
- Page 5, line 248: $dg(\rho)$ → $d\rho\,g(\rho)$ (twice)
- Page 5, Proposition 2, item 2: The expression $E(y,\tilde{b})$ does not match the definition (equation (4)) of the energy function $E(y)$.
- Page 6, line 284, equation (6), 413, 428: The second-order terms on the right-hand sides would need the prefactor $\frac{1}{2}$.
- Page 6, line 313: dynamic(s) is stable
- Page 7, line 342: $\sqrt{2(N-k)}$ would require a prefactor.
- Page 8, Proposition 5: The outermost parentheses on the right-hand side of the formula defining $E_1(u)$ are redundant and should be removed.
- Page 8, equation (11): $R^\top F+F^\top R$ → $R^\top F(u)+F(u)^\top R$; The outermost parentheses on the right-hand side of the formula defining $F(u)$ are redundant and should be removed.
- Page 8, lines 406-408, 411: $Wu-b$ → $Wu+b$ (four times)
- Page 8, line 410: gives $F(u)$ → gives $\frac{\partial E}{\partial u}=F(u)f(u)$; $\dot{u}=R(u)f(u(t))$ → $\dot{u}=-R(u)f(u(t))$
- Page 9, line 447: $E=E_1$ → $E=E_2$
- Page 9, Example 7: $erf$ should be in the upright face.
- Page 10, line 497: one observe(s)

Decision: Accept (Poster)