# GroupMamba: Parameter-Efficient and Accurate Group Visual State Space Model

Abdelrahman Shaker
Mohamed bin Zayed University of Artificial Intelligence
&Syed Talal Wasim
University of Bonn
&Salman Khan
Mohamed bin Zayed University of Artificial Intelligence
&Juergen Gall
University of Bonn
&Fahad Shahbaz Khan
Mohamed bin Zayed University of Artificial Intelligence; Linköping University

###### Abstract

Recent advancements in state-space models (SSMs) have showcased effective
performance in modeling long-range dependencies with subquadratic complexity.
However, pure SSM-based models still face challenges related to stability and
achieving optimal performance on computer vision tasks. Our paper addresses
the challenges of scaling SSM-based models for computer vision, particularly
the instability and inefficiency of large model sizes. To address this, we
introduce a Modulated Group Mamba layer which divides the input channels into
four groups and applies our proposed SSM-based efficient Visual Single
Selective Scanning (VSSS) block independently to each group, with each VSSS
block scanning in one of the four spatial directions. The Modulated Group
Mamba layer also wraps the four VSSS blocks into a channel modulation operator
to improve cross-channel communication. Furthermore, we introduce a
distillation-based training objective to stabilize the training of large
models, leading to consistent performance gains. Our comprehensive experiments
demonstrate the merits of the proposed contributions, leading to superior
performance over existing methods for image classification on ImageNet-1K,
object detection, instance segmentation on MS-COCO, and semantic segmentation
on ADE20K. Our tiny variant with 23M parameters achieves state-of-the-art
performance with a classification top-1 accuracy of 83.3% on ImageNet-1K,
while being 26% efficient in terms of parameters, compared to the best
existing Mamba design of same model size. Our code and models are available
at: https://github.com/Amshaker/GroupMamba.

##  1 Introduction

Various context modeling methods have emerged in the domains of language and
vision understanding. These include Convolution [22, 66], Attention [59], and,
more recently, State Space Models [17, 16]. Transformers with their multi-
headed self-attention mechanism [59] have been central to both language models
such as GPT-3 [2] and vision models such as Vision Transformers [10, 36].
However, challenges arose due to the quadratic computational complexity of
attention mechanisms particularly for longer sequences, leading to the recent
emergence of State Space models such as S4 [17].

While being effective in handling extended input sequences due to their linear
complexity in terms of sequence lengths, S4 [17] encountered limitations in
global context processing in information-dense data, especially in domains
like computer vision due to the data-independent nature of the model.
Alternatively, approaches such as global convolutions-based state space models
[14] and Liquid S4 [20] have been proposed to mitigate the aforementioned
limitations. The recent Mamba [16] introduces the S6 architecture which aims
to enhance the ability of state-space models to handle long-range dependencies
efficiently. The selective-scan algorithm introduced by Mamba uses input-
dependent state-space parameters, which allow for better in-context learning
while still being computationally efficient compared to self-attention.

However, Mamba, specifically the S6 algorithm, is known to be unstable for
e.g., image classification, especially when scaled to large sizes [45].
Additionally, the Mamba model variant used in image classification generally
called the VSS (Visual State Space) block, can be more efficient in terms of
parameters and compute requirements based on the number of channels. The VSS
block includes extensive input and output projections along with depth-wise
convolutions, whose parameters and compute complexities are directly
proportional to the number of channels in the input. To address this issue, we
propose a _Modulated Group Mamba_ layer which mitigates the aforementioned
issues in a computation and parameter-efficient manner. The main contributions
of our paper are:

  1. 1.

We introduce a _Modulated Group Mamba_ layer, inspired by Group Convolutions,
which enhances computational efficiency and interaction in state-space models
by using a multi-direction scanning method for comprehensive spatial coverage
and effective modeling of local and global information.

  2. 2.

We introduce a _Channel Affinity Modulation (CAM)_ operator, which enhances
communication across channels to improve feature aggregation, addressing the
limited interaction inherent in the grouping operation.

  3. 3.

We employ a distillation-based training objective to further stabilize the
training of models with a large number of parameters.

  4. 4.

We build a series of parameter-efficient generic classification models called
“GroupMamba”, based on the proposed _Modulated Group Mamba_ layer. Our tiny
variant achieves 83.3% top-1 accuracy on ImageNet-1k [9] with $23M$ parameters
and $4.6G$ FLOPs. Additionally, our base variant achieves top-1 accuracy of
84.5% with $57M$ parameters and $14G$ FLOPs, outperforming all recent SSM
methods (see Figure 1).

Figure 1: Left: Comparison in terms of Parameters vs. Top-1 Accuracy on
ImageNet-1k [9]. Our GroupMamba-B achieves superior top-1 classification
accuracy while reducing parameters by 36% compared to VMamba [34]. Right:
Qualitative results of GroupMamba-T on semantic segmentation (top right), and
object detection and instance segmentation (bottom right). More qualitative
examples are presented in Figure 3 and Figure 4

##  2 Related Work

Convolutional Neural Networks (ConvNets) have been the popular choice for
computer vision tasks since the introduction of AlexNet [29]. The field has
rapidly evolved with several landmark ConvNet architectures [51, 55, 22, 25,
56]. Alongside these architectural advances, significant efforts have been
made to refine individual convolution layers, including depthwise convolution
[65], group convolution [7], and deformable convolution [8]. Recently,
ConvNeXt variants [37, 63] have taken concrete steps towards modernizing
traditional 2D ConvNets by incorporating macro designs with advanced settings
and training recipes to achieve on-par performance with the state-of-the-art
models.

In recent years, the pioneering Vision Transformer (ViT) [10] has
significantly impacted the computer vision field, including tasks such as
image classification [57, 36, 35, 12], object detection [3, 71, 43, 68], and
segmentation [5, 50, 27]. ViT [10] introduces a monolithic design that
approaches an image as a series of flattened 2D patches without image-specific
inductive bias. The remarkable performance of ViT for computer vision tasks,
along with its scalability, has inspired numerous subsequent endeavors to
design better architectures. The early ViT-based models usually require large-
scale datasets (e.g., JFT-300M [54]) for pretraining. Later, DeiT [57]
proposes advanced training techniques in addition to integrating a
distillation token into the architecture, enabling effective training on
smaller datasets (e.g., ImageNet-1K [9]). Since then, subsequent studies have
designed hierarchical and hybrid architectures by combining CNN and ViT
modules to improve performance on different vision tasks [53, 40, 11, 49, 12].
Another line of work is to mitigate the quadratic complexity inherent in self-
attention, a primary bottleneck of ViTs. This effort has led to significant
improvements and more efficient and approximated variants [61, 49, 44, 42, 28,
6, 58], offering reduced complexity while maintaining effectiveness.

Recently, State Space Models (SSMs) have emerged as an alternative to ViTs
[59], capturing the intricate dynamics and inter-dependencies within language
sequences [17]. One notable method in this area is the structured state-space
sequence model (S4) [17], designed to tackle long-range dependencies while
maintaining linear complexity. Following this direction, several models have
been proposed, including S5 [52], H3 [13], and GSS [41]. More recently, Mamba
[16] introduces an input-dependent SSM layer and leverages a parallel
selective scan mechanism (S6).

In the visual domain, various works have applied SSMs to different tasks. In
particular for image classification, VMamba [34] uses Mamba with bidirectional
scans across both spatial dimensions in a hierarchical Swin-Transformer [36]
style design to build a global receptive field efficiently. A concurrent work,
Vision Mamba (Vim) [70], instead proposed a monolithic design with a single
bidirectional scan for the entire image, outperforming traditional vision
transformers like DeiT. LocalVMamba [26] addresses the challenge of capturing
detailed local information by introducing a scanning methodology within
distinct windows (inspired from Swin-Transformer [36]), coupled with dynamic
scanning directions across network layers. EfficientVMamba [46] integrates
atrous-based selective scanning and dual-pathway modules for efficient global
and local feature extraction, achieving competitive results with reduced
computational complexity. These models have been applied for image
classification, as well as image segmentation [33, 39, 48, 15], video
understanding [67, 30, 4], and various other tasks [19, 23, 60, 18, 31]. Their
wide applicability shows the effectiveness of SSMs [17, 52, 13, 41], and in
particular Mamba [16], in the visual domain. In this paper, we propose a
_Modulated Group Mamba_ layer that mitigates the drawbacks of the default
vision Mamba block, such as lack of stability [45] and the increased number of
parameters with respect to the number of channels.

##  3 Method

Figure 2: Overview of the proposed method. Top Row: The overall architecture
of our framework with a consistent hierarchical design comprising four stages.
Bottom Row: We present (b) The design of the modulated group mamba layer. The
input channels are divided into four groups with a single scanning direction
for each VSSS block. This significantly reduces the computational complexity
compared to the standard mamba layer, with similar performance. Channel
Affinity Modulation mechanism is introduced to address the limited
interactions within the VSSS blocks. (c) The design of VSSS block. It consists
of Mamba block with 1D Selective Scanning block followed by FFN. (d) The four
scanning directions used for the four VSSS blocks are illustrated.

Motivation: Our method is motivated based on the observations with respect to
the limitations of existing Visual State-Space models.

  * •

Lack of Stability for Larger Models: We observe from [45] that Mamba [16]
based image classification models with an MLP channel mixer are unstable when
scaled to a large number of parameters. This instability can be seen in
SiMBA-L (MLP) [45], which leads to sub-optimal classification results of
$49\%$ accuracy. We mitigate this issue by introducing a _Modulated Group
Mamba_ design alongside a distillation objective (as presented in Section 3.4)
that stabilizes the Mamba SSM training without modifying the channel mixer.

  * •

Efficient Improved Interaction: Given the computational impact of Mamba-based
design on the number of channels, the proposed _Modulated Group Mamba_ layer
is computationally inexpensive and parameter efficient than the default Mamba
and able to model both local and global information from the input tokens
through multi-direction scanning. An additional _Channel Affinity Modulation_
operator is proposed in this work to compensate for the limited channel
interaction due to the grouped operation.

###  3.1 Preliminaries

State-Space Models: State-space models (SSMs) like S4 [17] and Mamba [16] are
structured sequence architectures inspired by a combination of recurrent
neural networks (RNNs) and convolutional neural networks (CNNs), with linear
or near-linear scaling in sequence length. Derived from continuous systems,
SSMs define and 1D function-to-function map for an input
$x(t)\in\mathbb{R}^{L}\rightarrow y(t)\in\mathbb{R}^{L}$ via a hidden state
$h(t)\in\mathbb{R}^{N}$. More formally, SSMs are described by the continuous
time Ordinary Differential Equation (ODE) in Equation 1.

$\begin{split}h^{\prime}(t)&={\mathbf{A}}h(t)+{\mathbf{B}}x(t),\\\
y(t)&={\mathbf{C}}h(t),\end{split}$ (1)

where $h(t)$ is the current hidden state, $h^{\prime}(t)$ is the updated
hidden state, $x(t)$ is the current input, $y(t)$ is the output,
${\mathbf{A}}\in\mathbb{R}^{N\times N}$ is SSM’s evolution matrix, and
${\mathbf{B}}\in\mathbb{R}^{N\times 1},{\mathbf{C}}\in\mathbb{R}^{N\times 1}$
are the input and output projection matrices, respectively.

Discrete State-Space Models: To allow these models to be used in sequence
modeling tasks in deep learning, they need to be discretized, converting the
SSM from a continuous time function-to-function map into a discrete-time
sequence-to-sequence map. S4 [17] and Mamba [16] are among the discrete
adaptations of the continuous system, incorporating a timescale parameter
${\mathbf{\Delta}}$ to convert the continuous parameters
${\mathbf{A}},{\mathbf{B}}$ into their discrete equivalents
$\overline{{\mathbf{A}}},\overline{{\mathbf{B}}}$. This discretization is
typically done through the Zero-Order Hold (ZOH) method given in Equation 2.

$\begin{split}\overline{{\mathbf{A}}}&=\exp({\mathbf{\Delta}\mathbf{A}}),\\\
\overline{{\mathbf{B}}}&=({\mathbf{\Delta}\mathbf{A}})^{-1}(\exp({\mathbf{\Delta}\mathbf{A}})-{\mathbf{I}})\cdot{\mathbf{\Delta}\mathbf{B}}\\\
h_{t}&=\overline{{\mathbf{A}}}h_{t-1}+\overline{{\mathbf{B}}}x_{t},\\\
y_{t}&={\mathbf{C}}h_{t}.\end{split}$ (2)

While both S4 [17] and Mamba [16] utilize a similar discretization step as
stated above in Equation 2, Mamba differentiates itself from S4 by
conditioning the parameters ${\mathbf{\Delta}}\in\mathbb{R}^{B\times L\times
D}$, ${\mathbf{B}}\in\mathbb{R}^{B\times L\times N}$ and
${\mathbf{C}}\in\mathbb{R}^{B\times L\times N}$, on the input
$x\in\mathbb{R}^{B\times L\times D}$, through the S6 Selective Scan Mechanism,
where $B$ is the batch size, $L$ is the sequence length, and $D$ is the
feature dimension.

###  3.2 Overall Architecture

As shown in Figure 2 (a), our model uses a hierarchical architecture, similar
to Swin Transformer [36], with four stages to efficiently process images at
varying resolutions. Assuming an input image,
$\mathbf{I}\in\mathbb{R}^{H\times W\times 3}$, we first apply a Patch
Embedding layer to divide the image into non-overlapping patches of size
$4\times 4$ and embed each patch into a $C_{1}$-dimensional feature vector.
The patch embedding layer is implemented using two $3\times 3$ convolutions
with a stride of 2. This produces features maps of size
$\frac{H}{4}\times\frac{W}{4}\times C_{1}$ at the first stage. These feature
maps are passed to a stack of our Modulated Grouped Mamba blocks (as detailed
in Section 3.3). In each subsequent stage, a down-sampling layer merges
patches in a $2\times 2$ region, followed by another stack of our Modulated
Grouped Mamba blocks. Hence, feature size at stages two, three and four are
$\frac{H}{8}\times\frac{W}{8}\times C_{2}$,
$\frac{H}{16}\times\frac{W}{16}\times C_{3}$, and
$\frac{H}{32}\times\frac{W}{32}\times C_{4}$, respectively.

###  3.3 Modulated Group Mamba Layer

We present the overall operations of the proposed _Modulated Group Mamba_
layer (Figure 2 (b)) for an input sequence $\mathbf{X}_{\textsf{in}}$, with
dimensions $(B,H,W,C)$, where $B$ is the batch size, $C$ is the number of
input channels and $H$/$W$ are the width and height of the feature map, in
Equation 3.

$\displaystyle\mathbf{X}_{\textsf{GM}}$
$\displaystyle=\textsf{GroupedMamba}(\mathbf{X}_{\textsf{in}},\Theta)$ (3)
$\displaystyle\mathbf{X}_{\textsf{CAM}}$
$\displaystyle=\textsf{CAM}(\mathbf{X}_{\textsf{GM}},\textsf{Affinity}(\mathbf{X}_{\textsf{in}}))$
$\displaystyle\mathbf{X}_{\textsf{out}}$
$\displaystyle=\mathbf{X}_{\textsf{in}}+\textsf{FFN}(\textsf{LN}(\mathbf{X}_{\textsf{CAM}}))$

Here, $\mathbf{X}_{\textsf{GM}}$ is the output of Equation 6,
$\mathbf{X}_{\textsf{CAM}}$ is the output of Equation 9, LN is the Layer
Normalization [1] operation, FFN is the Feed-Forward Network as described by
Equation 5, and $\mathbf{X}_{\textsf{out}}$ is the final output of the
Modulated Group Mamba block. The individual operations, namely the
GroupedMamba operator, the VSSS block used inside the GroupedMamba operator,
and the CAM operator, are presented in Section 3.3.1, Section 3.3.2 and
Section 3.3.3, respectively.

####  3.3.1 Visual Single Selective Scan (VSSS) Block

The VSSS block (Figure 2 (c)) is a token and channel mixer based on the Mamba
operator. Mathematically, for an input token sequence
$\mathbf{Z}_{\textsf{in}}$, the VSSS block performs the operations as
described in Equation 4.

$\displaystyle\mathbf{Z}^{\prime}_{\textsf{out}}$
$\displaystyle=\mathbf{Z}_{\textsf{in}}+\textsf{Mamba}(\textsf{LN}(\mathbf{Z}_{\textsf{in}}))$
(4) $\displaystyle\mathbf{Z}_{\textsf{out}}$
$\displaystyle=\mathbf{Z}^{\prime}_{\textsf{out}}+\textsf{FFN}(\textsf{LN}(\mathbf{Z}^{\prime}_{\textsf{out}}))$

Where $\mathbf{Z}_{\textsf{out}}$ is the output sequence, Mamba is the
discretized version of the Mamba SSM operator as described in Equation 2.

$\textsf{FFN}(\textsf{LN}(\mathbf{Z}^{\prime}_{\textsf{out}}))=\textsf{GELU}(\textsf{LN}(\mathbf{Z}^{\prime}_{\textsf{out}})\mathbf{W}_{1}+\mathbf{b}_{1})\mathbf{W}_{2}+\mathbf{b}_{2}$
(5)

Where GELU [24] is the activation function and $\mathbf{W}_{1}$,
$\mathbf{W}_{2}$, $\mathbf{b}_{1}$, and $\mathbf{b}_{2}$ are weights and
biases for the linear projections.

####  3.3.2 Grouped Mamba Operator

Considering the motivation presented earlier in Section 3, we aim to design a
variant of the Mamba [16] that is both computationally efficient and can
effectively model the spatial dependencies of the input sequence. Given that
Mamba is computationally inefficient on large number of channels $C$ in the
input sequence, we propose a grouped variant of the operator, inspired by
Grouped Convolutions. The Grouped Mamba operation is a variant of the VSSS
block presented in Section 3.3.1, where the input channels are divided into
groups, and the VSSS operator is applied separately to each group.
Specifically, we divide the input channels into four groups, each of size
$\frac{C}{4}$, and an independent VSSS block is applied to each group. To
better model spatial dependencies in the input, each of the four groups scans
in one of four directions across the token sequence: left-to-right, right-to-
left, top-to-bottom, and bottom-to-top, as outlined in Figure 2 (d).

Let $G=4$ be the number of groups representing four scanning directions: left-
to-right, right-to-left, top-to-bottom, and bottom-to-top. We form four
sequences from the input sequence $\mathbf{X}_{\textsf{in}}$, namely
$\mathbf{X}_{\textsf{LR}}$, $\mathbf{X}_{\textsf{RL}}$,
$\mathbf{X}_{\textsf{TB}}$, and $\mathbf{X}_{\textsf{BT}}$, each of shape
$(B,H,W,\frac{C}{4})$, representing one of the four directions specified
earlier. These are then flattened to form a single token sequence of shape
$(B,N,\frac{C}{4})$, where $N=W\times H$ is the number of tokens in the
sequence. The parameters for each of the four groups can be specified by
$\theta_{\textsf{LR}}$, $\theta_{\textsf{RL}}$, $\theta_{\textsf{TB}}$, and
$\theta_{\textsf{BT}}$, respectively, for each of the four groups,
representing the parameters for the VSSS blocks.

Given the above definitions, the overall relation for the Grouped Mamba
operator can be written as shown in Equation 6.

$\displaystyle\begin{aligned}
\mathbf{X}_{\textsf{GM}}=\textsf{GroupedMamba}(\mathbf{X}_{\textsf{in}},\Theta)=\textsf{Concat}(&\textsf{VSSS}(\mathbf{X}_{\textsf{LR}},\Theta_{\textsf{LR}}),\\\
&\textsf{VSSS}(\mathbf{X}_{\textsf{RL}},\Theta_{\textsf{RL}}),\\\
&\textsf{VSSS}(\mathbf{X}_{\textsf{TB}},\Theta_{\textsf{TB}}),\\\
&\textsf{VSSS}(\mathbf{X}_{\textsf{BT}},\Theta_{\textsf{BT}}))\end{aligned}$
(6)

Where:

  * •

$\mathbf{X}_{\textsf{LR}}$, $\mathbf{X}_{\textsf{RL}}$,
$\mathbf{X}_{\textsf{TB}}$, and $\mathbf{X}_{\textsf{BT}}$ represent the input
tensors scanned in the respective directions.

  * •

$\Theta_{\textsf{LR}}$, $\Theta_{\textsf{RL}}$, $\Theta_{\textsf{TB}}$, and
$\Theta_{\textsf{BT}}$ represents the parameters of the VSSS block for each
direction.

  * •

The output of each Mamba operator is reshaped again to $(B,H,W,\frac{C}{4})$,
and concatenated back to form the token sequence $\mathbf{X}_{\textsf{GM}}$,
again of the size $(B,H,W,C)$.

####  3.3.3 Channel Affinity Modulation (CAM)

On its own, the Grouped Mamba operator may have a disadvantage in the form of
limited information exchange across channels, given the fact that each
operator in the group only operates over $\frac{C}{4}$ channels. To encourage
the exchange of information across channels, we propose a Channel Affinity
Modulation operator, which recalibrates channel-wise feature responses to
enhance the representation power of the network. In this block, we first
average pool the input to calculate the channel statistics as shown in
Equation 7.

$\textsf{ChannelStat}(\mathbf{X}_{\textsf{in}})=\textsf{AvgPool}(\mathbf{X}_{\textsf{in}})$
(7)

where $\mathbf{X}_{\textsf{in}}$ is the input tensor, and AvgPool represents
the global average pooling operation. Next comes the affinity calculation
operation as shown in Equation 8.

$\textsf{Affinity}(\mathbf{X}_{\textsf{in}})=\sigma\left(W_{2}\delta\left(W_{1}\textsf{ChannelStat}(\mathbf{X}_{\textsf{in}})\right)\right)$
(8)

where $\delta$ and $\sigma$ represent non-linearity functions, and $W_{1}$ and
$W_{2}$ are learnable weights. The role of $\sigma$ is to assign an importance
weight to each channel to compute the affinity. The result of the affinity
calculation is used to recalibrate the output of the Grouped Mamba operator,
as shown in Equation 9.

$\mathbf{X}_{\textsf{CAM}}=\textsf{CAM}(\mathbf{X}_{\textsf{GM}},\textsf{Affinity}(\mathbf{X}_{\textsf{in}}))=\mathbf{X}_{\textsf{GM}}\cdot\textsf{Affinity}(\mathbf{X}_{\textsf{in}})$
(9)

where $\mathbf{X}_{\textsf{CAM}}$ is the recalibrated output,
$\mathbf{X}_{\textsf{GM}}$ is the concatenated output of the four VSSS groups
from Equation 6, $\mathbf{X}_{\textsf{in}}$ is the input tensor, and
$\textsf{Affinity}(\mathbf{X}_{\textsf{in}})$ are the channel-wise attention
scores obtained from the channel affinity calculation operation in Equation 8.

###  3.4 Distilled Loss Function

As mentioned earlier in the motivation in Section 3, the Mamba training is
unstable when scaled to large models [45]. To mitigate this issue, we propose
to utilize a distillation objection alongside the standard cross-entropy
objective. Knowledge distillation involves training a student model to learn
from a teacher model’s behavior by minimizing a combination of the
classification loss and distillation loss. The distillation loss is computed
using the cross-entropy objective between the logits of the teacher and
student models. Given the logits ($Z_{s}$) from the student model, logits
($Z_{t}$) from a teacher model (RegNetY-16G [47] in our case), the ground
truth label $y$, and the hard decision of the teacher
$y_{\mathrm{t}}=\mathrm{argmax}_{c}Z_{\mathrm{t}}(c)$, the joint loss function
is defined as shown in Equation 10.

$\mathcal{L}_{\mathrm{total}}=\alpha\mathcal{L}_{\mathrm{CE}}(Z_{s},y)+(1-\alpha)\mathcal{L}_{\mathrm{CE}}(Z_{s},y_{\mathrm{t}}).$
(10)

where $\mathcal{L}_{\mathrm{CE}}$ is the cross-entropy objective and $\alpha$
is the weighting parameter. We experimentally show in Section 4 that training
with this distillation objective stabilizes training, leading to consistent
performance gains on larger model variants.

##  4 Experiments

###  4.1 Image Classification

Settings: The image classification experiments are based on ImageNet-1K [9],
which comprising of over $1.28$ million training images and 50K validation
images, spanning $1,000$ categories. Following [35], we train our models for
using the AdamW [38] optimizer and a cosine decay learning rate scheduler for
$300$ epochs, including a $20$ epoch warm-up. The total batch size is set to
$1024$, with models trained on 8x A100 GPUs, each with 80GB of CUDA memory.
Optimizer betas are set to $(0.9,0.999)$; momentum is set to $0.9$, and an
initial learning rate of $1\times 10^{-3}$ is used with a weight decay of
$0.05$. Label smoothing of $0.1$ is used alongside the distillation objective
(see Section 3.4).

Results: Table 1 presents a comparison of our proposed GroupMamba models (T,
S, B) with various state-of-the-art methods. The GroupMamba models exhibit a
notable balance of accuracy and computational efficiency. GroupMamba-T
achieves a top-1 accuracy of $83.3\%$ with $23$ million parameters and $4.5$
GFLOPs, outperforming ConvNeXt-T [37] and Swin-T [36] by $1.2\%$ and $2.0\%$,
respectively, with fewer parameters. Additionally, GroupMamba-T surpasses the
recently introduced SSM models, outperforming VMamba-T [34] and LocalVMamba-T
[26] by $0.8\%$ and $0.6\%$, respectively, while using $26\%$ fewer parameters
than VMamba-T. GroupMamba-S, with $34$ million parameters and $7.0$ GFLOPs,
achieves an accuracy of $83.9\%$, surpassing VMamba-S [34], Swin-S [36], and
EfficientVMamba-B [46]. The performance is better than LocalVMamba-S [26] by
0.2% with $32\%$ fewer parameters. Furthermore, GroupMamba-B achieves an
accuracy of $84.5\%$ with only $57$ million parameters and $14$ GFLOPs,
exceeding VMamba-B [34] by $0.6\%$ while using $36\%$ fewer parameters.

Table 1: Performance comparison of GroupMamba models with state-of-the-art convolution-based, attention-based, and SSM-based models on ImageNet-1K [9]. Our models demonstrate superior performance and achieve a better trade-off between accuracy and model parameters. Method |  | Token
---
mixing
| Image
---
size
#Param. | FLOPs | Top-1 acc.
RegNetY-8G [47] | Conv | 2242 | 39M | 8.0G | 81.7
RegNetY-16G [47] | Conv | 2242 | 84M | 16.0G | 82.9
EffNet-B4 [56] | Conv | 3802 | 19M | 4.2G | 82.9
EffNet-B5 [56] | Conv | 4562 | 30M | 9.9G | 83.6
EffNet-B6 [56] | Conv | 5282 | 43M | 19.0G | 84.0
DeiT-S [57] | Attention | 2242 | 22M | 4.6G | 79.8
DeiT-B [57] | Attention | 2242 | 86M | 17.5G | 81.8
DeiT-B [57] | Attention | 3842 | 86M | 55.4G | 83.1
ConvNeXt-T [37] | Conv | 2242 | 29M | 4.5G | 82.1
ConvNeXt-S [37] | Conv | 2242 | 50M | 8.7G | 83.1
ConvNeXt-B [37] | Conv | 2242 | 89M | 15.4G | 83.8
Swin-T [36] | Attention | 2242 | 28M | 4.6G | 81.3
Swin-S [36] | Attention | 2242 | 50M | 8.7G | 83.0
Swin-B [36] | Attention | 2242 | 88M | 15.4G | 83.5
ViM-S [70] | SSM | 2242 | 26M | - | 80.5
VMamba-T [34] | SSM | 2242 | 31M | 4.9G | 82.5
VMamba-S [34] | SSM | 2242 | 50M | 8.7G | 83.6
VMamba-B [34] | SSM | 2242 | 89M | 15.4G | 83.9
LocalVMamba-T [26] | SSM | 2242 | 26M | 5.7G | 82.7
LocalVMamba-S [26] | SSM | 2242 | 50M | 11.4G | 83.7
EfficientVMamba-B [46] | SSM | 2242 | 33M | 4.0G | 81.8
GroupMamba-T | SSM | 2242 | 23M | 4.6G | 83.3
GroupMamba-S | SSM | 2242 | 34M | 7.0G | 83.9
GroupMamba-B | SSM | 2242 | 57M | 14G | 84.5

###  4.2 Object Detection and Instance Segmentation

Settings: We evaluate the performance of GroupMamba-T for object detection on
the MS-COCO 2017 dataset [32]. Our method is based on the Mask-RCNN [21]
detector with the hyperparameters as used for Swin [36]. We use the AdamW [38]
optimizer and train Mask-RCNN with GroupMamba-T backbone for $12$ epochs. The
backbone is initialized and fine-tuned from the ImageNet-1K [9]. We use an
initial learning rate of $1\times 10^{-4}$ and decay by a factor of $10$ at
epochs $9$ and $11$.

Results: Table 2 shows the results of GroupMamba-T, comparing it against
various state-of-the-art models for object detection and instance segmentation
using the Mask R-CNN framework on the MS-COCO dataset. Our model achieves box
AP (AP${}^{\text{b}}$) of 47.6 and mask AP (AP${}^{\text{m}}$) of 42.9. It
surpasses ResNet-50 [22], Swin-T [35], ConvNeXt-T [37]. In addition,
GroupMamba-T has competitive performance compared to VMamba-T [34] and
LocalVMamba-T [26], with less 20% parameters compared to VMamba-T. In Figure
3, we present the qualitative results of GroupMamba-T on samples from the COCO
validation set [32], demonstrating its performance in instance segmentation
and object detection. Our model accurately localizes objects and correctly
segments them across diverse scenes and varying scales.

###  4.3 Semantic Segmentation

Settings: We also evaluate the performance of GroupMamba-T for semantic
segmentation on the ADE20K [69] dataset. The framework is based on the UperNet
[64] architecture, and we follow the same hyperparameters as used for the Swin
[36] backbone. More specifically, we use the AdamW [38] optimizer for a total
of $160k$ iterations with an initial learning rate of $6\times 10^{-5}$. The
default input resolution used in our experiments is $512\times 512$.

Results: The GroupMamba-T model demonstrates favorable performance in semantic
segmentation compared to various state-of-the-art methods, as presented in
Table 3. GroupMamba-T achieves a mIoU of $48.6$ in single-scale and $49.2$ in
multi-scale evaluation, with $49M$ parameters and $955G$ FLOPs. This
outperforms ResNet-50 [22], Swin-T [36], and ConvNeXt-T [37]. Additionally,
GroupMamba-T exceeds the performance of the recent SSM methods, including
ViM-S [70], VMamba-T [34], and LocalVMamba [26] with fewer number of
parameters. In Figure 4, we show qualitative results of GroupMamba-T on
samples from the ADE20K [69] validation set for semantic segmentation. The
first row shows the ground truth masks, while the second row displays the
predicted masks. It is notable that our model delineates the masks accurately,
highlighting the effectiveness of semantic segmentation. The quantitative and
qualitative results of GroupMamba demonstrate the robust generalization
capability of our GroupMamba backbones across diverse downstream tasks,
including semantic segmentation, object detection, and instance segmentation.

Table 2: Performance comparison for object detection and instance segmentation
on MS-COCO [32] using Mask R-CNN [21]: $AP^{b}$ and $AP^{m}$ signify box AP
and mask AP, respectively. FLOPs, are computed for an input dimension of
$1280\times 800$. Mask R-CNN 1$\times$ schedule
---
Backbone | AP${}^{\text{b}}$ | AP${}^{\text{b}}_{\text{50}}$ | AP${}^{\text{b}}_{\text{75}}$ | AP${}^{\text{m}}$ | AP${}^{\text{m}}_{\text{50}}$ | AP${}^{\text{m}}_{\text{75}}$ | #param. | FLOPs
ResNet-50 [22] | 38.2 | 58.8 | 41.4 | 34.7 | 55.7 | 37.2 | 44M | 260G
Swin-T [36] | 42.7 | 65.2 | 46.8 | 39.3 | 62.2 | 42.2 | 48M | 267G
ConvNeXt-T [37] | 44.2 | 66.6 | 48.3 | 40.1 | 63.3 | 42.8 | 48M | 262G
PVTv2-B2 [62] | 45.3 | 67.1 | 49.6 | 41.2 | 64.2 | 44.4 | 45M | 309G
VMamba-T [34] | 47.4 | 69.5 | 52.0 | 42.7 | 66.3 | 46.0 | 50M | 270G
LocalVMamba-T [26] | 46.7 | 68.7 | 50.8 | 42.2 | 65.7 | 45.5 | 45M | 291G
GroupMamba-T | 47.6 | 69.8 | 52.1 | 42.9 | 66.5 | 46.3 | 40M | 279G
Figure 3: Qualitative results of GroupMamba-T for object detection and instance segmentation on the COCO validation set. Table 3: Performance comparison for semantic segmentation on ADE20K [69] using UperNet [64]. The terms ’SS’ and ’MS’ refer to evaluations conducted at single-scale and multi-scale levels, respectively. FLOPs are computed for an input dimension of $512\times 2048$. method | crop size | mIoU (SS) | mIoU (MS) | #param. | FLOPs
---|---|---|---|---|---
ResNet-50 [22] | $512^{2}$ | 42.1 | 42.8 | 67M | 953G
DeiT-S + MLN | $512^{2}$ | 43.8 | 45.1 | 58M | 1217G
Swin-T [36] | $512^{2}$ | 44.4 | 45.8 | 60M | 945G
ConvNeXt-T [37] | $512^{2}$ | 46.0 | 46.7 | 60M | 939G
ViM-S [70] | $512^{2}$ | 44.9 | - | 46M | -
VMamba-T [34] | $512^{2}$ | 48.3 | 48.6 | 62M | 948G
EfficientVMamba-B [46] | $512^{2}$ | 46.5 | 47.3 | 65M | 930G
LocalVMamba-T [26] | $512^{2}$ | 47.9 | 49.1 | 57M | 970G
GroupMamba-T | $512^{2}$ | 48.6 | 49.2 | 49M | 955G

###  4.4 Ablation Study

Table 4 illustrates the effect of each of our contributions, compared to
VMamba-T. The proposed GroupMamba-T without the CAM module achieves a top-1
accuracy of 82.2% in the second row. Integrating the proposed CAM module, as
shown in the third row, improves performance by 0.3%, with comparable
parameters and FLOPs. In the fourth row, we expand the number of channels to
match the parameter count of VMamba-T (without using distillation). Our
GroupMamba-T* surpasses VMamba-T by 0.6% with an equivalent number of
parameters. In the final row, we incorporate the proposed distillation loss
without expanding the channels, which boosts performance by 0.8% compared to
VMamba-T while using 26% fewer parameters.

Figure 4: Qualitative results of GroupMamba-T for semantic segmentation on ADE20K validation set. The first row shows the ground truth for the masks, while the second and second show the corresponding predictions of our model. Table 4: Ablation study on GroupMamba-T for the proposed CAM module and Distilled Loss. Method | #Param. | FLOPs | Top-1 acc.
---|---|---|---
VMamba-T [34] | 31M | 4.9G | 82.5
GroupMamba-T w/o CAM | 22M | 4.6G | 82.2
GroupMamba-T with CAM | 23M | 4.6G | 82.5
GroupMamba-T* with CAM | 31M | 5.4G | 83.1
GroupMamba-T with CAM + Distilled Loss | 23M | 4.6G | 83.3

##  5 Conclusion

In this paper, we tackle the computational inefficiencies and stability
challenges associated with visual SSMs for computer vision tasks by
introducing a novel layer called the _Modulated Group Mamba_. We also propose
a multi-directional scanning method that improves parameter efficiency by
scanning in four spatial directions and leveraging the _Channel Affinity
Modulation_ (CAM) operator to enhance feature aggregation across channels. To
stabilize training, especially for larger models, we employ a distillation-
based training objective. Our experimental results demonstrate that the
proposed GroupMamba models outperform recent SSMs while requiring fewer
parameters.

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# Long-wavelength fluctuations and dimensionality crossover in confined
liquids

Jing Yang  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Yan-Wei Li  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Massimo Pica Ciamarra<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore 637371, Singapore
CNR–SPIN, Dipartimento di Scienze Fisiche, Università di Napoli Federico II,
I-80126, Napoli, Italy

###### Abstract

The phase behavior of liquids confined in a slit geometry does not reveal a
crossover from a three- to a two-dimensional behavior as the gap size
decreases. Indeed, the prototypical two-dimensional hexatic phase only occurs
in liquids confined to a monolayer. Here, we demonstrate that the
dimensionality crossover is apparent in the lateral size dependence of the
relaxation dynamics of confined liquids, developing a Debye model for the
density of vibrational states of confined systems and performing extensive
numerical simulations. In confined systems, Mermin-Wagner fluctuations enhance
the amplitude of vibrational motion or Debye-Waller factor by a quantity
scaling as the inverse gap width and proportional to the logarithm of the
aspect ratio, as a clear signature of a two-dimensional behaviour. As the
temperature or lateral system size increases, the crossover to a size-
independent relaxation dynamics occurs when structural relaxation takes place
before the vibrational modes with the longest wavelength develop.

##  I Introduction

The phase behaviour and dynamics of liquids confined in slit geometries are
affected by the competition of several length scales. Indeed, for a liquid
confined in a slit of dimension $L\times L\times H$, the lateral length $L$
and gap width $H\ll L$ interfere with bulk-liquid length scales, such as the
typical distance between the particles, $a_{0}=\rho^{-1/3}$, and the
structural correlation length, $\xi_{\rm bulk}\simeq 10$, e.g., as estimated
from the decay of the radial distribution function [1, 2, 3]. The competition
between $H$ and $\xi_{\rm bulk}$ induces a cascade of confinement-induced
ordering transitions [4, 5, 6, 2], and a solid like behaviour interpreted as a
signal of a first-order transition [7, 8] or, more recently [9, 10, 11], as a
continuous glass transition. For molecular liquids in very narrow
confinements, length scales associated with the anisotropic molecular
structure [1, 12, 13, 14] and the details of the interaction between the
molecules and the confining walls also play a role.

The rich and system-dependent phase behaviour of confined systems makes
difficult rationalizing the crossover from three to two dimensions focusing on
its gap size dependence. Indeed the hexatic phase, which is a phase with
short-ranged translational order and long-ranged bond-orientational order only
occurring in two-dimensional systems, has been only reported for $H\simeq
a_{0}$ in Lennard-Jones systems [15]. In this extremely confined limit, the
occurrence of a two-dimensional behaviour is in line with the observed
decoupling of the lateral and transverse degrees of freedom [16, 17].

The size dependence of the relaxation dynamics of confined liquids offers an
alternative and unexplored approach to investigate the dimensionality
crossover. Indeed, two-dimensional systems differ from their three-dimensional
counterpart because Mermin-Wagner [18] long-wavelength (LW) fluctuations make
their relaxation dynamics size dependent [19, 20, 21, 22, 23, 24, 25]. This
alternative approach is also convenient as Mermin-Wagner fluctuations are
always present in two-dimensional systems; conversely, the two- and the three-
dimensional phase behaviour do not qualitatively differ in all systems [26,
27, 28].

In this paper, we demonstrate that confined systems have a relaxation dynamics
depending on the lateral size $L$, as two-dimensional ones, and rationalize
the dimensionality crossover clarifying how this $L$ dependence varies with
the gap width $H$ and relaxation time. We find that, in the solid regime,
confinement enhances the asymptotic value of the mean-square displacement, or
Debye-Waller factor, by a factor scaling as $(1/H)\ln(L/H)$. A similar
enhancement of the mean square displacement occurs in the liquid phase.
Liquids, however, exhibit a dimensionality crossover as size-effects vanish
above a characteristic $H$-independent system size fixed by sound velocity and
relaxation time. We further clarify that our predictions apply to both
molecular and colloidal liquids through the investigation of experimentally
relevant confinement settings.

##  II Debye’s DOS in confinement

We develop a Debye-like model for the vibrational density of states (DOS) of
confined amorphous solids to rationalize the size dependence of their
dynamical properties. In confinement, the length scales $L$ and $H$ and the
transverse sound velocity $c_{s}$ fix two characteristic frequencies,
$\omega_{\rm L}=2\pi c_{s}/L$ and $\omega_{\rm H}=2\pi c_{s}/H$. $\omega_{\rm
L}$ is the smallest possible phonon frequency. The physical role of
$\omega_{\rm H}$ is understood considering that phonons with
$\omega<\omega_{\rm H}$, which have a wavelength larger than $H$, do not fit
along the transverse direction. Hence $\omega_{\rm H}$ separates the spectrum
into a low-frequency region, $\omega_{\rm L}<\omega<\omega_{\rm H}$ where
excitations are essentially two dimensional, and in a high frequency region,
$\omega_{\rm H}<\omega<\omega_{\rm D}$, with $\omega_{\rm D}$ the Debye’
frequency, where excitations are three dimensional. In the Debye’
approximation, the density of states is

$D(\omega)=\left\\{\begin{aligned} &c~{}\frac{\omega}{\omega_{\rm
D}^{2}}&&{\omega_{\rm L}\leq\omega\leq\omega_{\rm H}}\\\
&c~{}\frac{\omega^{2}}{\omega_{\rm H}\omega_{\rm D}^{2}}&&{\omega_{\rm
H}\leq\omega\leq\omega_{\rm D}},\end{aligned}\right.$ (1)

with $c$ non-dimensional normalization constant,

$c^{-1}=\frac{1}{2}\left[\left(\frac{\omega_{\rm H}}{\omega_{\rm
D}}\right)^{2}-\left(\frac{\omega_{\rm L}}{\omega_{\rm
D}}\right)^{2}\right]-\frac{1}{3}\left[\frac{\omega_{\rm D}}{\omega_{\rm
H}}-\left(\frac{\omega_{\rm H}}{\omega_{\rm D}}\right)^{3}\right].$ (2)

$D(\omega)$ is schematically illustrated in Fig. 1(a). We remark that we have
restricted the above investigation to the transverse modes, which are of
greater relevance to our purposes as having a smaller frequency. The
longitudinal modes can be similarly described.

The vibrational density of states allows us to evaluate the asymptotic value
of the mean square displacement, or the Debye-Waller factor, averaging the
contributions $k_{B}T/m\omega^{2}$ of the different modes. To highlight the
dependence on the different length scales involved, we write $\omega_{\rm
D}=2\pi c_{s}/\lambda_{\rm D}$, finding

${\rm DW}=\frac{k_{B}T}{m\omega_{\rm
D}^{2}}\frac{\ln\left(\frac{L}{H}\right)+\frac{H}{\lambda_{\rm
D}}-1}{\frac{1}{2}\left[\left(\frac{\lambda_{\rm
D}}{H}\right)^{2}-\left(\frac{\lambda_{\rm
D}}{L}\right)^{2}\right]+\frac{1}{3}\left[\frac{H}{\lambda_{\rm
D}}-\left(\frac{\lambda_{\rm D}}{H}\right)^{2}\right]}.$ (3)

The three dimensional limit, ${\rm DW_{3D}}\simeq\frac{3k_{B}T}{m\omega_{\rm
D}^{2}}$, and the two-dimensional one, ${\rm
DW_{2D}}=\frac{2k_{B}T}{m\omega_{\rm D}^{2}}\ln\left(\frac{L}{H}\right)$, are
recovered for $H\to L\gg\lambda_{\rm D}$ and for $H\to\lambda_{\rm D}\ll L$,
respectively.

In quasi-2D systems, $L\gg H\gg\lambda_{D}$, Eq. 3 is approximated by

${\rm DW}\simeq{\rm DW_{3D}}\left[1+\frac{\lambda_{\rm
D}}{H}\left(\ln\left(\frac{L}{H}\right)-1\right)\right].$ (4)

Hence, we predict that in confined systems the DW grows logarithmically with
$L$, as in 2D, with a slope decreasing as $1/H$. We remark here that, as long
as $H\gg\lambda_{\rm D}$, the DW factor grows as $H$ decreases at constant
$L$, e.g., as the system becomes more confined. This occurs because, as $H$
decreases, a larger fraction of the phonon spectrum becomes effectively two-
dimensional.

##  III Numerical details

We validate our theoretical prediction, and explore the effect of confinement
on the liquid phase, via extensive molecular dynamics simulations [29] of the
standard A:B 80:20 Kob-Andersen (KA) Lennard-Jones (LJ) mixture [30], where
particles interact via the potential
$V_{\alpha\beta}\left({r}\right)=4\epsilon_{\alpha\beta}\left[\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{12}-\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{6}+{C}_{\alpha\beta}\right]$,
and $\epsilon_{AB}=1.5\epsilon_{AA}$, $\epsilon_{BB}=0.5\epsilon_{AA}$,
$\sigma_{AB}=0.8\sigma_{AA}$, $\sigma_{BB}=0.88\sigma_{AA}$,
$\alpha,\beta\in\left\\{{A,B}\right\\}$. The potential is truncated at
$r_{c}=2.5\sigma_{\alpha\beta}$, and $C_{\alpha\beta}$ enforces $V(r_{c})=0$.
The mass of the particles $m$, $\epsilon_{AA}$, and $\sigma_{AA}$ are our unit
of mass, energy and distance, respectively. We first thermalize the system in
the NPT ensemble, at $P=1.0$, allowing the box size to vary only in the
lateral dimensions. Production runs are then performed in the NVE ensemble.
The number of particles depends on $L$ and $H$, and varies between $10^{3}$ to
$~{}10^{6}$ million. We average the dynamical data over at least four
independent runs.

We monitor the relaxation dynamics studying the mean square displacement,
$\langle\Delta r^{2}(t)\rangle=\frac{1}{N}\sum\Delta\mathbf{r}_{i}^{2}(t)$,
where $\Delta\mathbf{r}_{i}$ is the displacement of particle $i$ at time $t$,
and the self-scattering function,
$F_{s}\left(k,t\right)=\frac{1}{N}\left<\sum_{j=1}^{N}e^{i\mathbf{k}\cdot\Delta\mathbf{r}_{j}\left(t\right)}\right>$,
where $\mathbf{k}$ the wavevector of the first peak of the static structure
factor of bulk systems. The relaxation time $\tau$ is defined by
$F_{s}\left(k,\tau\right)=1/e$. We further investigate the dynamics using the
cage-relative mean square displacement and self-scattering function [20, 21,
22, 31]. These are defined as above, with the displacement of particle $i$
replaced by its cage-relative counterpart, $\Delta_{\rm
CR}\mathbf{r}_{i}=\Delta\mathbf{r}_{i}-\frac{1}{n_{i}}\sum_{j}\Delta\mathbf{r}_{j}$,
where the sum is over all neighbors of particle $i$ at time $t=0$. We identify
the neighbors via the Voronoi construction.

Figure 1:  (a) Schematic illustration of the Debye’s density of states of
quasi-2D systems, Eq. 1. (b) Low-frequency cumulative density of states of
confined solids with lateral length $L=80$ and different gap sizes $H$. (c)
The data in ${\bf a}$ collapses when plotted vs. $\omega H$ and vertically
scaled, for $H>\xi_{\rm bulk}\simeq 10$. (d) Mean square displacement at
$T=0.005$ and $H=10$, for different $L$ values. (e) The asymptotic DW factor
grows logarithmically with the lateral size $L$, with a slope scaling as $1/H$
(inset). Errors are smaller than the symbol size.  Figure 2:  The dependence
of the average density on the gap width, at $T=0.35$, when periodic boundary
conditions are used in the confining direction. Confinement does not strongly
influence the average density, in the range of gap widths we have considered.
The radial distribution function, shown in the inset for $H=20$ and $L=20$ at
$T=0.35$, approaches one at $\xi_{\rm bulk}/2\simeq 5$.

We consider three different confinement approaches. First, we use periodic
boundary conditions in the confining direction, which is an approach that is
useful to avoid layering as well as to compare with the theoretical
predictions. When using this approach, the density is essentially constant,
$\rho=1.1775(5)$, as we illustrate in Fig. 2(a). In the figure we also show
that, for larger $H$ values representative of the bulk limit, the radial
distribution function becomes constant for $r\simeq\xi_{\rm bulk}/2\simeq 5$.

Secondly, we confine the system between flat walls. In this case, the
interaction between particles of type $i=A,B$ and the walls is given by a LJ
potential with energy scale $\epsilon_{ii}$ and length scale $\sigma_{ii}$,
truncated in its minimum. In the presence of flat walls, the density sensibly
decreases with $H$, and layering occurs, as shown in Fig. 2b.

Finally, we perform simulations of systems confined between rough walls. In
this case, we first thermalize at the desired state pressure large samples,
using periodic boundary conditions in all directions, and then freeze the
positions of all particles whose height is outside the interval $[0:H]$. When
using rough walls, we work at fixed density rather than at fixed pressure.

##  IV Confined amorphous solids

We study the density of states of confined amorphous solid configurations
generated by minimizing the energy of configurations equilibrated at low
temperature. We fix the pressure of these low-temperature configurations to
$P=1$ by adjusting the lateral size, which slightly fluctuates around $L=80$.
We considered several $H$ values, so that the number of particles ranges from
$36000$ to $150000$. We further use periodic boundary conditions in all
spatial directions to prevent structural inhomogeneities due to layering,
hence allowing for a more transparent comparison with the theoretical
predictions. The effect of walls is discussed in Sec. VI.

Figure 3:  Long-wavelength fluctuations in confined amorphous solids. (a) Mean
square displacement, and (b), self-scattering function, at three different
values of the temperature. We fix $H=10$ and show, at each temperature,
results for $10\leq L\leq 320$. (c) The relaxation time decreases as the
lateral size increases, while the cage-relative relaxation time is
$L$-independent. (d) The relaxation time decreases as the gap-size decreases,
particularly for $H\leq\xi_{\rm bulk}$, while the cage-relative relaxation
time is $H$-independent. The relaxation times in (c) and (d) are divided by
their respective values at $L=10$ and at $H=30$, to facilitate their
comparison. In (c) and (d), errors are smaller than the symbol size.

We evaluate the low-frequency end of the vibrational spectrum of the generated
energy minima via the direct diagonalization of their Hessian matrix. To
compare the numerical results with our theoretical prediction of Eq. 1,
schematically illustrated in Fig. 1(a), we focus on the frequency dependence
of the cumulative distribution $C(\omega)=\int D(\omega)d\omega$. Due to the
large lateral size of our systems [32], we observe gaps at low frequency, as
predicted by linear elasticity 111We have verified that these gaps are not an
artefact of the discontinuity of the force at the cutoff distance [39], as
they persist when the interaction potential is appropriately smoothed.. Figire
1(b) also demonstrates that $C(\omega)/\omega^{2}$ is constant at small
frequencies, and increases above an $H$ dependent crossover frequency which,
according to Eq. 1, should scale as $\omega_{H}\propto c_{s}/H$. Indeed, when
plotted versus $\omega H$, and vertically scaled, the data collapse up to
their crossover point, as we illustrate in Fig. 1(c). The figure also supports
the $\omega^{2}$ to $\omega^{3}$ crossover for the cumulative distribution
suggested by the theoretical model.

We remark that the data collapse of Fig. 1(c) breaks for small $H$. To
rationalize this observation, we investigate in Fig. 2 the gap size dependence
of the density and the radial correlation function of a low-temperature solid
configuration. We observe that the density is almost $H$ independent, for
$H\geq 5$, and that the radial correlation function approaches the ideal gas
limit at $r\simeq 5$. This allows us to estimate the structural correlation
length of the bulk solid, $\xi_{\rm bulk}\simeq 10$. We thus understand that,
in Fig. 1(c), no collapse occurs for small $H$ as confinement interferes with
the structural correlation length of the system.

We further validate our theoretical prediction for the dependence of the DW
factor of amorphous solids on the relevant length scales $L$ and $H$, Eq. 4,
performing simulations at a low-temperature value at which structural
relaxation is negligible. In this limit, the mean-square displacement
approaches a constant DW value at long times, as illustrated in Fig. 1(d) for
$H=10$. Figure 1(e) shows that this limiting DW factor grows as the logarithm
of the lateral size $L$, with a slope scaling as $1/H$, in agreement with the
predictions of Eq. 4.

Figure 4:  Dimensionality crossover in confined liquids. (a) The mean square
displacement exhibits a crossover between two different regimes at a time
$t_{\rm LW}\simeq 0.3L$ (dash-dotted line). Dashed lines are polynomial fits
used to estimate the mean square displacement at the crossover time (circles).
Data are for $T=0.35$, and different $L$ values. (b) The mean square
displacement at the crossover time grows faster that $\ln L$ (open symbols),
above a characteristic $T$ dependent lateral system size. When this occurs,
structural relaxation rather than LWs dominate the diffusivity, and hence the
system has a 3D-like behaviour. (c) State points with an effective two-
dimensional behaviour according to the analysis in (b), are illustrated as
open circles. Diamonds, conversely, identify those having a three-dimensional
behaviour. Stars correspond to the prediction of Eq. 5, $L=\alpha
c_{s}\tau_{CR}(T)$, with $\alpha\simeq 0.018$. The interpolating solid line is
a guide to the eye. All panels refer to $H=10$. Supplemental Fig. S4 shows
that the results are insensitive to changes in the gap width.

##  V Confined liquids

Having ascertained that LWs influence the behaviour of confined solids, we now
demonstrate that they similarly affect the relaxation dynamics of quasi-2D
supercooled-liquids. To this end, we investigate the size and temperature
dependence of the mean square displacement and self-scattering function at the
wave vector of the peak of the static structure factor of bulk systems.
Figures 3(a) and (b) show that the transient solid-like response revealed by
the mean square displacement and the self-scattering function becomes less
apparent as the system size decreases. This size dependence is more apparent
at low temperature, where the transient solid like behaviour is manifest.

We prove that this observed size dependence originates from LW fluctuations by
comparing the $L$ dependence of the relaxation time $\tau$ and of the cage-
relative (CR) relaxation time $\tau_{\rm CR}$. Cage-relative quantities,
indeed, are insensitive to collective particle displacements and hence filter
out the effect of LWs [20, 21, 22]. In Fig. 3(c), we observe that, while the
standard relaxation time decreases logarithmically with $L$, the CR one is $L$
independent. These results closely parallel those observed in strictly two-
dimensional systems [19, 20, 21, 22, 23, 24, 25] and demonstrate that LW
fluctuations sensibly affect the structural relaxation dynamics of confined
liquids.

In Fig. 3(d), we further show that the relaxation time $\tau$ decreases as the
gap width is reduced and a larger fraction of the vibrational spectrum becomes
effectively two-dimensional. This dynamical speed up is particularly relevant
for $H<\xi_{\rm bulk}$, indicating that the structural changes induced by such
strong confinement promote LW fluctuations. This is consistent with the
observation of a significant increment in the density of low-frequency modes
for $H=5$, in Fig. 1(b). The gap independence of the cage-relative relaxation
time, also illustrated in Fig. 3(d), confirms our interpretation, namely that
the $H$-induced speed-up originates from LW fluctuations.

Figure 5:  Long-wavelength fluctuations in slit geometries. (a) The transverse
mean-square-displacement and (b) the self-intermediate scattering function for
supercooled liquids with various transverse length scales $10\leq L\leq 320$
at the same perpendicular length scale $H=10$. (c) Width dependence of the
relaxation time, and of the cage-relative relaxation time for a $L=40$ system.
Errors are smaller than the symbol size. The inset is a schematic diagram of
the confining geometry.

We quantitatively investigate the dimensionality crossover focusing on the
mean square displacement, $\langle\Delta r^{2}(t)\rangle$. In the solid phase,
$\langle\Delta r^{2}(t)\rangle$ approaches an asymptotic DW factor value on a
time scale $t_{\rm LW}\propto\omega_{L}^{-1}\propto L$. The asymptotic value
of the DW factor grows as $\ln L/f(L)$, with $f(L)$ a slowing increasing
function of $L$, corresponding to the denominator of Eq. 3. In the liquid
phase, therefore, we expect a crossover in the time dependence of the mean
square displacement at a time $t_{\rm LW}$. Figures 4(a) and 4(b) demonstrate
that such a crossover occurs at $t_{\rm LW}\simeq 0.3L$, for $T=0.38$. At the
same $t_{\rm LW}$ similar crossovers occur at all temperatures.

When LW fluctuations dominate the dynamics, as in the solid phase,
$\frac{\langle\Delta r^{2}(t_{\rm LW})}{\ln L}\rangle\propto 1/f(L)$ decreases
with $L$. We therefore assume LW fluctuations to become negligible at $L$
values at which $\langle\Delta r^{2}(t_{\rm LW})\rangle$ grows faster than
$\ln L$. When this occurs, irreversible relaxation events rather than large-
amplitude oscillations dominate the diffusivity. In Fig. 4(b) we indeed
observe that $\langle\Delta r^{2}(t_{\rm LW})\rangle/\ln L$ is not monotonic
in $L$, decreasing with $L$ when LWs are relevant (solid symbols), and
increasing when they are not (open symbols). This behavior allows us to
identify crossover $L$ values, which we have verified not to depend on the gap
width. This study leads to the $L$-$T$ diagram of Fig. 4(c). The system-size
dependent dynamics characteristic of two-dimensional behaviour occurs at low
temperature and small lateral size and disappears as either the lateral length
or the temperature increase. We remark that while this diagram does not depend
on the confinement width $H$, size effects gradually fade away as $1/H$, as in
the solid phase, and hence become not appreciable at large $H$.

We exploit the size independence of the cage-relative relaxation time to
rationalize this observed dimensionality crossover. Indeed, vibrational
excitations cannot last more than the cage-relative relaxation time, as on
this time scale the structure of the system sensibly changes, as particles
change neighbours. Since the vibrational modes influencing the structural
relaxation dynamics are those that have time to develop, we expect the
crossover between a two-dimensional size-dependent relaxation dynamics and a
three-dimensional size-independent relaxation dynamics to occur at

$\frac{L}{\tau_{\rm CR}(T)}=\alpha c_{s},$ (5)

with $c_{s}$ being the transverse sound velocity and $\alpha$ being a
constant. In other words, size effects disappear for $L>\alpha c_{s}\tau_{\rm
CR}(T)$, as the system relaxes before the lowest size-dependent mode develops.
This theoretical prediction well describes the data of Fig. 4(d).

##  VI Effect of smooth and rough walls

Our theoretical analysis and numerical simulations demonstrate that LW
fluctuations affect the dynamics of confined liquids. However, so far we have
described simulations obtained using periodic boundary conditions in all
spatial directions; one might wonder, therefore, whether LWs also play a role
in the experimentally relevant set up of liquids confined between two parallel
walls at a separation $H$. To address this question, we investigate the
relaxation dynamics of the KA LJ binary mixture confined between two
atomically-smooth flat walls. Since the walls prevent diffusion along the
transverse direction, we focus on particle motion in the lateral directions,
effectively defining two-dimensional mean-square displacement and self-
scattering function. We find that, under wall confinement, the relaxation
dynamics has the typical size dependence induced by LW fluctuations, the
caging regime becoming less apparent as $L$ increases, as we illustrate in
Figs. 5(a) and 5(b).

Figure 6:  Dependence of the average density on the gap width, at $T=0.35$,
for a system confined in between flat walls at a separation $H$. The density
decreases as the gap width decreases. Flat walls, furthermore, induce
layering, as we illustrate in the inset by plotting the density at a distance
$h$ from a confining wall.  Figure 7:  Mean-square displacement (a) and self-
scattering function (b) of systems confined between rough walls, at $H=2$ and
$H=4$. These quantities are evaluated focusing on the behavior of the central
layer of particles. The relaxation dynamics does not depend on the lateral
length, or system size $N$, indicating that the rough walls kill the LW
fluctuations.

The structural changes induced by the walls, however, strongly affect the
relaxation dynamics, as evidenced by the $H$ dependence of the standard and CR
relaxation times, which we illustrate in Fig. 5(c). For $H\geq\xi_{\rm bulk}$,
both relaxation times decrease as the system becomes more confined; this is,
we believe, the combined effect of layering and the reduction in the average
density induced by the confinement, which we illustrate in Fig. 6.

Importantly, we observe in Fig. 5(c) that for $H\leq\xi_{\rm bulk}$, while the
relaxation time decreases as the gap width is reduced, the cage-relative
relaxation sharply increases. This increase in the CR-relaxation time is in
qualitative agreement with the many previous investigations reporting an
increase in the viscosity of molecular liquids under confinement [1, 34, 35,
9, 11]. Indeed, we remind the reader that viscosity and cage-relative
relaxation time are related [36, 25]. This observed decoupling demonstrates
that smooth walls do not kill the LWs, but rather make their effect more
apparent.

While smooth walls do not kill LWs, rough walls strongly suppress them.
Indeed, we show in Fig. 7 that the relaxation dynamics of liquids confined
between rough walls does not depend on the lateral system size. We remark that
for very large gap widths the effect of the boundary should become negligible,
and hence LW fluctuations should play a role. Since the influence of LWs on
the dynamics scales as $1/H$, however, their effect in this large-$H$ limit
may be not easily appreciated. We expect variations [13] in the roughness of
the confining walls and the wall-liquid interaction potential to only
qualitatively affect the observed phenomenology.

##  VII Conclusions and experimental relevance

The confinement-induced enhancement of the DW factor described Eq. 3 is an
equilibrium property not affected by the underlying microscopic dynamics,
equally valid for molecular and colloidal solids. In the supercooled regime,
the signatures of LW fluctuations conversely depend on how much the system
moves along the phase-space directions of the low-frequency modes before
particles rearrange. Since the size of this displacement depends on the
microscopic dynamics and it is smaller if the system moves diffusively, rather
than ballistically, we expect the influence of confinement to be more relevant
at the molecular scale rather than at the colloidal scale. Nevertheless, we
remind the reader that LWs are observed in experiments [22, 25, 23] and
simulations [25] of two-dimensional colloidal systems; our predictions
concerning the role of LW fluctuations in confined systems therefore apply to
both molecular and colloidal systems.

For the effect of LW fluctuations to be experimentally visible, however, the
roughness scale of the confining walls must be smaller than the size of the
particles. Rough walls, indeed, affect the motion in the lateral dimensions
and kill the LW fluctuations, as we have shown in Fig. 7. The requirement of
smooth confining walls is not a technical limitation. Walls that are de facto
flat at the molecular scale exist [10], and it is undoubtedly possible to
confine large colloidal particles between walls that are flat at the particle
scale. In colloidal experiments, however, one should ascertain that no
particles stick irreversibly to the walls, effectively making them rough,
e.g., as observed in Refs. [37, 38]. Hence our predictions are experimentally
testable both in confined molecular liquids, e.g., comparing the size
dependence of the viscosity and of structural relaxation time, and in confined
colloidal systems, comparing, e.g. the standard and cage-relative relaxation
times.

Our results show that confined systems exhibit a gradual dimensionality
crossover controlled by the gap width and the temperature, which is
appreciable when investigating the lateral size dependence of the dynamics.
The physics of confined liquids is thus richer than previously realised. These
findings might be relevant to a variety of applications involving micro- and
nanofluidics, e.g., lab-on-a-chip devices, where particles flow in confined
geometries.

###### Acknowledgements.

We acknowledge support from the Singapore Ministry of Education through the
Academic Research Fund Tier RG 86/19(S), Singapore, and the National
Supercomputing Centre Singapore (NSCC) for the computational resources.

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