# Label Distribution Learning via Implicit Distribution Representation

Zhuoran Zheng1 and Xiuyi Jia1
1CSE, Nanjing University of Science and Technology Corresponding authors.

###### Abstract

In contrast to multi-label learning, label distribution learning characterizes
the polysemy of examples by a label distribution to represent richer
semantics. In the learning process of label distribution, the training data is
collected mainly by manual annotation or label enhancement algorithms to
generate label distribution. Unfortunately, the complexity of the manual
annotation task or the inaccuracy of the label enhancement algorithm leads to
noise and uncertainty in the label distribution training set. To alleviate
this problem, we introduce the implicit distribution in the label distribution
learning framework to characterize the uncertainty of each label value.
Specifically, we use deep implicit representation learning to construct a
label distribution matrix with Gaussian prior constraints, where each row
component corresponds to the distribution estimate of each label value, and
this row component is constrained by a prior Gaussian distribution to moderate
the noise and uncertainty interference of the label distribution dataset.
Finally, each row component of the label distribution matrix is transformed
into a standard label distribution form by using the self-attention algorithm.
In addition, some approaches with regularization characteristics are conducted
in the training phase to improve the performance of the model.

##  1 Introduction

Label distribution learning (LDL) (Geng (2016)) is a novel learning paradigm
that characterizes the polysemy of examples. In LDL, the relevance of each
label to an example is given by an exact numerical value between 0 and 1 (also
known as description degree), and the description degree of all labels forms a
distribution to fully characterize the polysemy of an example. Compared with
traditional learning paradigms, LDL is a more generalizable and
representational learning paradigm that provides richer semantic information.

LDL has been successful in several application domains (Gao et al. (2018);
Zhao et al. (2021); Chen et al. (2021a); Si et al. (2022)). To obtain the
label distribution for learning, there are mainly two ways: one is expert
labeling, but labeling is expensive and there is no objective labeling
criterion, and the resulting label distribution is highly subjective and
ambiguous. The other is to convert a multi-label dataset into a label
distribution dataset through a label enhancement algorithm (Xu et al. (2019;
2020); Zheng et al. (2021a); Zhao et al. (2022)). However, label enhancement
lacks a reliable theory to ensure that the label distribution recovered from
logical labels converges to the true label distribution, because logical
labels provide a very loose solution space for the label distribution, making
the solution less stable and less accurate.

In summary, the label distribution dataset used for training has a high
probability of inaccuracy and uncertainty, which significantly limits the
performance of LDL algorithms. To characterize and mitigate the uncertainty of
the label distribution, we propose a novel LDL method based on the implicit
label distribution representation. Our work is inspired by recent work on
implicit neural representation in 2D image reconstruction (Sitzmann et al.
(2020)). The key idea of implicit neural representation is to represent an
object as a function that maps a sequence of coordinates to the corresponding
signal, where the function is de-parameterized by a deep neural network. In
this paper, we start with a deep network to extract the latent features of
input information. Then, the latent features are looked up against the encoded
coordinate matrix to generate a label distribution matrix (implicit
distribution representation). Finally, the label distribution matrix is
computed by a self-attention module to yield a standard label distribution.
Note that the goal of the proposed implicit distribution representation is to
generate a label distribution matrix with Gaussian distribution constraints as
a customized representation pattern.

To efficiently generate latent features around the coordinate, we design a
deep spiking neural network (Yamazaki et al. (2022)) with an MLP as an
executor to extract latent features in the input information. The architecture
of the whole network consists of multiple layers of linear spiking neurons,
and the neurons of different layers conduct a shortcut between them. Notably,
spiking neural networks have two key properties that are different from the
representation of artificial neural networks. First, a standard spiking neural
network considers the time characteristics $T$ (taking the image as an
example, the input tensor $X\in$ ${\mathbb{R}}^{T\times C\times W\times H}$.)
at a multi-step inference mechanism. Here, we create several pseudo-feature
spaces on the native feature space by setting different strategies with data
augmentation (Ucar et al. (2021)). These pseudo-features and the native
feature are stacked in the time dimension $T$ to achieve the multi-step
inference mechanism. Second, the representation capability of the spiking
neural network is underpowered, since the output space is a binarized sequence
({0 $\cdots$ 1}). Therefore, we place a standard MLP in the last layer of the
spiking neural network, which projects the features into the real number
space. Our model saves about 30$\sim$40$\%$ of energy consumption over ANNs
with the same network structure on embedded devices such as lynxi HP300,
Raspberry Pi, or Apple smartphones (PyTorch 1.2 support M1 Mac).

[capbesideposition=right,top,capbesidewidth=5.3cm]figure[]

Figure 1: Our architecture. This figure shows the architecture of the proposed
deep implicit function, which consists of two parts. The first part starts
with a latent feature prediction stream (SNN with an MLP) that learns the
input information to predict the feature maps. The second part learns a label
distribution matrix to regress a label distribution.

The extraction of latent features provides material for the coordinates to
generate the label distribution matrix. First, the initialized coordinate
matrix (the size is $L\times 64$, where $L$ denotes the number of labels, and
64 denotes features of the nodes) is reconstructed by using a GCN. Note that
the features of the nodes meet the Gaussian distribution since the deep
network needs to reconstruct the data from a fixed distribution. Then the
coordinate matrix is repeated in $N$ copies, and $N$ denotes the number of
samples. Next, the coordinate matrix computes a label distribution matrix (the
size is $N\times L\times 2L$) in the latent feature space by looking up the
table111https://pytorch.org/tutorials/intermediate/spatial_transformer_tutorial.html.
Each component of the label distribution matrix represents the distribution of
each label value, and the components are constrained by a priori Gaussian
distributions. Finally, the label distribution matrix leverages a self-
attention mechanism to obtain the corresponding label distribution of the
samples.

Yet, deep learning-based approaches are prone to overfitting manually
extracted features. To alleviate the problem, we propose some regularization
approaches to boost the performance of the model, and a new dataset based on
the image comprehension task is released. Our contribution includes: (i) For
LDL, this is a novel method to obtain the label distribution of a sample
through the implicit distribution representation. (ii) Spiking neural network
with an MLP is developed to save energy consumption of mobile devices, and
correlations between labels are deeply mined by a graph convolutional network.
(iii) Facing the LDL task, some regularization techniques are designed to
boost the performance of the model and a new LDL dataset is released.

##  2 Proposed Method

As shown in Figure 1, our architecture is a two-stage scheme, which is based
on an implicit representation approach. In the first stage (latent feature
extraction), the raw features $\mathcal{X}$ are fed into the encoder for
encoding to construct a latent feature space $F_{f}\in$ ${\mathbb{R}}^{L\times
H\times W}$. In the second stage (label distribution matrix learning), the
coordinate tensor (coordinate matrix is reshaped after GCN) $C_{r}\in$
${\mathbb{R}}^{L\times L\times 2)}$ is looked up in the latent feature space
$F_{f}$ to generate a label distribution matrix $\mathcal{M}\in$
${\mathbb{R}}^{L\times 2L}$, and finally a label distribution $\mathcal{D}$ is
obtained by using self-attention algorithm. Furthermore, some regularization
terms are introduced to our model to boost its performance.

Latent Feature Extraction. Our model starts with a latent feature extraction
task, which is based on a spiking neural network. Spiking neural networks have
a high potential value as a 3rd deep model and are efficient and
interpretable. In the LDL task, we develop a non-fixed Network Architecture
and a Network Implementation due to the diversity of LDL dataset formats.

Network Architecture: Our network consists of 17 layers of function units,
which include 8 linear units, 8 nonlinear units, and a transformation layer
(including an MLP, a mean operator, and a reshape operator). Essentially, this
is a residual network and the transformation layer at the tail of the network.
Except for the first layer (the number of neurons is the dimensionality of the
input features), each linear unit contains 1024 neurons, followed closely by a
ReLU non-linearity, and the last layer is a reshape function to generate the
feature space $F_{f}$. On datasets with a smaller feature space $\mathcal{X}$
(e.g., Gene dataset has only 36 features), each linear layer includes 64
neurons. We also attempted to use other activation functions than ReLU, such
as PReLU, Swish, and Sigmoid, but without any advantage. Besides, since the
spiking network involves a time dimension, the time dimension $T$ is squeezed
in the output space of the spiking network by using a mean operator. The
network’s tailgate should notice that the feature map is reshaped as $F_{f}$
using a reshape operator and the MLP (the number of neurons is $L\times
W\times H$.), where $H$ and $W$ are 32.

Network Implementations: The simplified residual network is implemented as an
ANN in two frameworks: PyTorch 1.12 (a standard deep network training library)
and SpikingJelly 0.0.13 (a library for deep learning applications that is part
of the PyTorch ecosystem). The SpikingJelly implementation can be run in
spiking or non-spiking modes. To train the spiking network, we use an ANN-SNN
conversion training approach. The conversion is simple, we convert the trained
model on PyTorch with the help of an ann2snn.Converter (SpikingJelly) to
obtain an SNN model. Note that the conversion method defines six modes (max,
$99.9\%$, $1.0\backslash 2$, $1.0\backslash 3$, $1.0\backslash 4$,
$1.0\backslash 5$) to obtain SNN with different accuracy, and in this paper,
we chose $99.9\%$. The literature (Patel et al. (2021)) provides a theoretical
basis for analyzing the conversion of ANN to SNN.

Label Distribution Matrix Learning. Inspired by confident learning (Northcutt
et al. (2021)), we develop a matrix of label distribution with a Gaussian
before estimating the uncertainty of the labels. For the data distribution of
each label value, we use a Gaussian distribution to delimit the distribution
rather than other multi-peaked distribution priors, and numerous kinds of
literature have verified that this approach can eliminate the uncertainty (Liu
et al. (2021); Zheng et al. (2021b); Ghosh et al. (2021); Li et al. (2022)).
Furthermore, to capture the global correlation between labels to generate a
standard label distribution, we employ a self-attention mechanism to model the
label distribution matrix. To obtain a label distribution matrix
$\mathcal{M}$, we need a latent feature $F_{f}$ from the SNN and a coordinate
matrix $C_{b}:\text{GCN}(C_{r})$ (GCN denotes the graph convolution network)
that passes through the graph convolution network. Specifically, first, we
initialize a matrix of coordinates $C_{r}\in$ ${\mathbb{R}}^{L\times 64}$
based on the functions (torch.randn) provided by PyTorch. $L$ denotes the
number of nodes and $64$ denotes that we assign one feature vector to each
node, with each feature vector sampled in a Gaussian distribution. To build a
graph structure the data needs information about the edges, where there is an
edge with no direction between the nodes. So far, we built the node with the
edge information and have the ability to integrate it into a graph to input
into the GCN. Our GCN includes four graph convolution layers and four
activation layers, where the activation function uses ReLU. These four graph
convolutions include 64, 128, 256, and $L\times L\times 2$ neurons
respectively. There is one key message to note, the output matrix $C_{b}$ of
the GCN is filled (torch.repeat) with the same number of samples as the latent
feature space $F_{f}$. Then, we use $C_{b}$ to look up the table in $F_{f}$ to
obtain a matrix by using flatten operations. The $L$ label distribution values
include a $1\times 2L$ vector to form a label distribution matrix
$\mathcal{M}$. Each vector is constrained by a designated Gaussian
distribution $\bar{\mathcal{M}}$ with parameters whose mean is the value of
the label distribution and variance of 0.5. Finally, this matrix is squeezed
by using a self-attention algorithm (Vaswani et al. (2017)) to obtain the
corresponding label distribution for the samples.

Regularization Techniques. We propose two techniques (Linear Normalization
Function, Data Augmentation) to boost the performance of our model.

Linear Normalization Function: Currently, most existing LDL algorithms use
Softmax to output a vector in the tail of the model.

$\text{Softmax}(d_{i}^{y_{i}})=\frac{\exp^{d_{i}^{y_{i}}}}{\sum_{c=1}^{C}\exp^{d_{c}^{y_{c}}}},$
(1)

where $d_{i}^{y_{i}}$ denotes a label value in the vector and $C$ denotes the
number of elements. Since the complexity of exponential operations is much
higher than linear operations, especially in the training phase of the model
(e.g. Gamma correction (Ju et al. (2019))). Several algorithms (such as
cosFormer (Qin et al. (2022))) seek to replace exponential operations. Here,
we develop a linear function Lnf to replace Softmax to output a label
distribution vector. Then predicted label distribution
$\widehat{\mathbf{d}_{i}}=$
$\left\\{\widehat{d_{i}^{y_{1}}},\widehat{d_{i}^{y_{2}}},\ldots,\widehat{d_{i}^{y_{c}}}\right\\}$
can be obtained by:

$\text{Lnf}(\widehat{d_{i}^{y_{i}}})=\frac{d_{i}^{y_{i}}+|\mathcal{D}_{\text{min}}|}{\sum_{c=1}^{C}(d_{c}^{y_{c}}+|\mathcal{D}_{\text{min}}|)},$
(2)

where $|\mathcal{D}_{\text{min}}|$ denotes the absolute value of the minimum
of the predicted label distribution values. Essentially, our method offsets
the values of the label distribution to the positive domain of the x-axis. We
evaluate both Softmax and linear normalization function on 12 datasets, and
our network with the linear normalization function converges 2.1 $\times$
faster than the model with Softmax.

Data Augmentation: Most LDL datasets are difficult to augment the samples with
expert knowledge since the input features are extracted manually. To overcome
this challenge, we introduce a simple and data-agnostic data augmentation
routine, termed mixup (Zhang et al. (2018)). In a nutshell, mixup constructs
virtual training examples:

$\displaystyle\tilde{x}=\lambda x_{i}+(1-\lambda)x_{j},\quad\text{ where
}x_{i},x_{j}\text{ are raw input vectors }$ (3)
$\displaystyle\tilde{y}=\lambda y_{i}+(1-\lambda)y_{j},\quad\text{ where
}y_{i},y_{j}\text{ are label distribution vectors }$

$\left(x_{i},y_{i}\right)$ and $\left(x_{j},y_{j}\right)$ are two examples
drawn at random from our training data, and $\lambda\in[0,1]$. However, a deep
network may still be over-fitted during the training phase, which leads to
poor generalization. We use the random masking scheme endowed on mixup,
formally expressed as:

$\displaystyle\tilde{x}=\lambda
x_{i}\times\text{mask}+(1-\lambda)x_{j}\times\text{mask},\quad\text{ where
}x_{i},x_{j}\text{ are raw input vectors }$ (4)
$\displaystyle\tilde{y}=\text{Lnf}(\lambda y_{i}+(1-\lambda)y_{j}),\quad\text{
where }y_{i},y_{j}\text{ are label distribution vectors }$

the mask is a vector represented by 1 or 0. In this paper, a mask contains 80%
of the scalar 1 and the rest is 0. Moreover, to solidify the definition of the
label distribution, we use Lnf to normalize the synthesized label vectors. We
also tried using other regularization schemes, such as random cropping, and
the results do not improve significantly.

Loss Function.

We optimize the weights and biases of the proposed network by minimizing the
$L_{1}$, KullbackLeibler (K-L) divergence, and regularization of the label
distribution matrix on the training set,

$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}\left\|\widehat{d_{i}}-d_{i}\right\|+\lambda\mathcal{L}_{kl}+\beta\sum_{i=1}^{N}\sum_{i=1}^{L}\mathcal{L}_{2}(\mathcal{M}_{ii},\bar{\mathcal{M}_{ii}}),$
(5)

where $N$ is the number of training samples, $d_{i}$ is the label distribution
vector by our model, and $\widehat{d}$ is the corresponding ground truth. The
weight $\lambda$ of the loss term $\mathcal{L}_{kl}$ (K-L) is set to 0.01 in
our experiments. In addition, the weight $\beta$ of the loss term (label
distribution matrix) is set to 0.1 in our experiments. Since it is difficult
to capture useful information by encoding a small number of tokens, encoding
for sequences of long tokens can extract higher-order semantics, such as
perceptual loss (Johnson et al. (2016)). Therefor, we used the above-mentioned
learning strategy on the datasets with several labels less than 20. On the
datasets with the number of labels greater than 20, we used $L_{1}$, K-L
divergence, perceptual loss, and regularization of the label distribution
matrix on the training set,

$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}\left\|\widehat{d_{i}}-d_{i}\right\|+\lambda_{1}\mathcal{L}_{kl}+\lambda_{2}\mathcal{L}_{p}+\beta\sum_{i=1}^{N}\sum_{i=1}^{L}\mathcal{L}_{2}(\mathcal{M}_{ii},\bar{\mathcal{M}_{ii}}),$
(6)

we use MLPs as the pre-trained model for the perceptual loss function
($\mathcal{L}_{p}$). The weight $\lambda_{1}$, $\lambda_{2}$, and $\beta$ are
set to 0.01, 0.01, 0.08 in our experiments, respectively. These MLPs consist
of three linear layers with fixed neurons and three activation layers (ReLU),
where the number of neurons is the same as the number of labels, using Kaiming
initialization (He et al. (2015)).

##  3 Experiments

Algorithm Configurations.

We conduct experiments on 12 datasets, the characteristics of the datasets are
reported in Table 1. Except for dataset wc-LDL, the configuration of all other
datasets is referenced to (Wang & Geng (2021)). This new release dataset (wc-
LDL) has 500 watercolor images and corresponding label distribution (12
emotions). wc-LDL is constructed with a thorough discussion in the
supplementary material. We develop the SNN with configurations on different
datasets also summarized in Table 1. To evaluate the performance of LDL
models, we use the six metrics proposed by (Geng (2016)), including Chebyshev
distance $\downarrow$, Clark distance $\downarrow$, Canberra distance
$\downarrow$, KL divergence $\downarrow$, Cosine similarity $\uparrow$, and
Intersection similarity $\uparrow$. LF and DA denote the loss function and
data augmentation method, respectively. $\downarrow$ represents the
indicator’s performance favoring low values and $\uparrow$ represents the
indicator’s performance favoring high values. $\mathscr{L}_{1,2:end}$ denotes
the number of neurons in the head layer and the rest of the layers of the SNN.

Table 1: Statistics of the experimental datasets with models. ID | Dataset | Examples | Features | Labels | Number of neurons of the SNN | LF | DA
---|---|---|---|---|---|---|---
1 | wc-LDL | 500 | 243 | 12 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [243, 1024] | Eq.5 | ✓
2 | SJAFFE | 213 | 243 | 6 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [243, 1024] | Eq.5 | ✓
3 | SBU-3DFE | 2500 | 243 | 6 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [243, 1024] | Eq.5 | ✗
4 | Scene | 2000 | 294 | 9 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [294, 1024] | Eq.5 | ✗
5 | Gene | 17892 | 36 | 68 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [36, 64] | Eq.6 | ✗
6 | Movie | 7755 | 1869 | 5 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [1869, 1024] | Eq.5 | ✗
7 | M2B | 1240 | 250 | 5 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [256, 1024] | Eq.5 | ✓
8 | SCUT | 1500 | 300 | 5 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [300, 1024] | Eq.5 | ✓
9 | fbp5500 | 5500 | 512 | 5 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [512, 1024] | Eq.5 | ✗
10 | RAF-ML | 4908 | 200 | 6 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [200, 1024] | Eq.5 | ✗
11 | Twitter | 10040 | 200 | 8 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [200, 1024] | Eq.5 | ✗
12 | Flickr | 11150 | 200 | 8 |  $\mathscr{L}_{1,2:end}$ $\rightarrow$ [200, 1024] | Eq.5 | ✗

Experimental Setting.

We conduct comparative experiments with five LDL algorithms (BFGS-LLD (Geng
(2016)), LDL-LRR (Jia et al. (2021a)), LDL-LCLR (Ren et al. (2019b)), LDLSF
(Ren et al. (2019a)) and LALOT (Zhao & Zhou (2018))) which also used data
augmentation schemes on 12 datasets. BFGS-LLD is based on a linear model, the
loss function is K-L divergence, and the optimization method is the quasi-
Newton approach. LDL-LRR and LDL-LCLR both consider label correlations in the
learning process, with the former considering the order relationship of the
labels and the latter capturing global relationships between labels. For LDL-
LRR, the parameters $\lambda$ and $\beta$ are selected from
$10^{\\{-6,-5,\ldots,-2,-1\\}}$ and $10^{\\{-3,-2,\ldots,1,2\\}}$,
respectively. For LDL-LCLR, the parameters
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ and $k$ are set to
$0.0001,0.001,0.001,0.001$ and 4, respectively. LDLSF leverages label-specific
features and common features simultaneously, whose parameters
$\lambda_{1},\lambda_{2}$ and $\lambda_{3}$ are selected from
$10^{\\{-6,-5,\ldots,-2,-1\\}}$, respectively, and $\rho$ is set to $10^{-3}$.
LALOT adopts optimal transport distance as the loss function, the trade-off
parameter $C$ and the regularization coefficient $\lambda$ is set to $200$ and
$0.2$, respectively. Our approach for experimental settings is reported in
Table 2. It is worth noting that Early stopping and Greed soup (Wortsman et
al. (2022)) are also used on all the datasets where the comparison algorithm
is executed.

Table 2: Training configuration of our model. ID | Dataset | Batch size | Epoch | Learning rate | Weight decay | Early stopping | Greed soup
---|---|---|---|---|---|---|---
1 | wc-LDL | 500 | 200 | $2\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
2 | SJAFFE | 213 | 200 | $2\times 10^{-2}$ | $1\times 10^{-4}$ | ✓ | ✓
3 | SBU-3DFE | 1000 | 200 | $1\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
4 | Scene | 1000 | 120 | $1\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
5 | Gene | 5000 | 150 | $2\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
6 | Movie | 2000 | 100 | $2\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
7 | M2B | 500 | 150 | $2\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
8 | SCUT | 500 | 150 | $1\times 10^{-2}$ | $1\times 10^{-4}$ | ✓ | ✓
9 | fbp5500 | 1500 | 300 | $2\times 10^{-2}$ | $1\times 10^{-4}$ | ✓ | ✓
10 | RAF-ML | 2000 | 100 | $1\times 10^{-3}$ | $1\times 10^{-4}$ | ✓ | ✓
11 | Twitter | 5000 | 200 | $1\times 10^{-3}$ | $1\times 10^{-4}$ | ✗ | ✗
12 | Flickr | 5000 | 200 | $1\times 10^{-2}$ | $1\times 10^{-4}$ | ✓ | ✓
Table 3: The performance of our proposed method with the comparison algorithms on 12 datasets. All algorithms enforce techniques such as early stopping, data augmentation, and greedy-soup to evaluate 12 datasets on the GPU. Our method is marked as gray. Dataset | Algorithm | Chebyshev $\downarrow$ | Clark $\downarrow$ | Canberra $\downarrow$ | K-L $\downarrow$ | Cosine $\uparrow$ | Intersection $\uparrow$
---|---|---|---|---|---|---|---
| Ours | 0.0779 $\pm$ 0.0021 | 0.3980 $\pm$ 0.0051 | 0.7779 $\pm$ 0.0030 | 0.04040 $\pm$ 0.0020 | 0.9883 $\pm$ 0.0009 | 0.8778 $\pm$ 0.0014
| LDL-LRR | 0.1122 $\pm$ 0.0030 | 0.4772 $\pm$ 0.0036 | 0.8802 $\pm$ 0.0024 | 0.05533 $\pm$ 0.0049 | 0.9510 $\pm$ 0.0022 | 0.8555 $\pm$ 0.0047
| LDL-LCLR | 0.1057 $\pm$ 0.0019 | 1.0569 $\pm$ 0.0039 | 0.7890 $\pm$ 0.0039 | 0.05045 $\pm$ 0.0037 | 0.9668 $\pm$ 0.0049 | 0.8383 $\pm$ 0.0018
| LDLSF | 0.1009 $\pm$ 0.0038 | 0.4199 $\pm$ 0.0044 | 0.9008 $\pm$ 0.0015 | 0.05199 $\pm$ 0.0040 | 0.9779 $\pm$ 0.0018 | 0.8660 $\pm$ 0.0022
| LALOT | 0.0989 $\pm$ 0.0019 | 0.6689 $\pm$ 0.0019 | 0.8089 $\pm$ 0.0049 | 0.04778 $\pm$ 0.0018 | 0.9476 $\pm$ 0.0020 | 0.8700 $\pm$ 0.0033
wc-LDL | BFGS-LLD | 0.1229 $\pm$ 0.0039 | 1.5657 $\pm$ 0.0021 | 0.7998 $\pm$ 0.0020 | 0.04998 $\pm$ 0.0051 | 0.9704 $\pm$ 0.0036 | 0.8611 $\pm$ 0.0016
| Ours | 0.0854 $\pm$ 0.0018 | 0.4008 $\pm$ 0.0030 | 0.7955 $\pm$ 0.0023 | 0.04100 $\pm$ 0.0012 | 0.9799 $\pm$ 0.0014 | 0.8809 $\pm$ 0.0015
| LDL-LRR | 0.1122 $\pm$ 0.0030 | 0.4772 $\pm$ 0.0036 | 0.8802 $\pm$ 0.0024 | 0.05533 $\pm$ 0.0049 | 0.9510 $\pm$ 0.0022 | 0.8555 $\pm$ 0.0047
| LDL-LCLR | 0.1057 $\pm$ 0.0019 | 1.0569 $\pm$ 0.0039 | 0.7890 $\pm$ 0.0039 | 0.05045 $\pm$ 0.0037 | 0.9668 $\pm$ 0.0049 | 0.8383 $\pm$ 0.0018
| LDLSF | 0.1009 $\pm$ 0.0038 | 0.4199 $\pm$ 0.0044 | 0.9008 $\pm$ 0.0015 | 0.05199 $\pm$ 0.0040 | 0.9779 $\pm$ 0.0018 | 0.8660 $\pm$ 0.0022
| LALOT | 0.0989 $\pm$ 0.0019 | 0.6689 $\pm$ 0.0019 | 0.8089 $\pm$ 0.0049 | 0.04778 $\pm$ 0.0018 | 0.9476 $\pm$ 0.0020 | 0.8700 $\pm$ 0.0033
SJAFFE | BFGS-LLD | 0.1229 $\pm$ 0.0039 | 1.5657 $\pm$ 0.0021 | 0.7998 $\pm$ 0.0020 | 0.04998 $\pm$ 0.0051 | 0.9711 $\pm$ 0.0036 | 0.8611 $\pm$ 0.0016
| Ours | 0.0833 $\pm$ 0.0020 | 0.3994 $\pm$ 0.0010 | 0.7611 $\pm$ 0.0020 | 0.03650 $\pm$ 0.0014 | 0.9811 $\pm$ 0.0015 | 0.8900 $\pm$ 0.0017
| LDL-LRR | 0.1109 $\pm$ 0.0036 | 0.4477 $\pm$ 0.0039 | 0.8666 $\pm$ 0.0026 | 0.05344 $\pm$ 0.0028 | 0.9597 $\pm$ 0.0029 | 0.8592 $\pm$ 0.0033
| LDL-LCLR | 0.1100 $\pm$ 0.0025 | 0.9660 $\pm$ 0.0039 | 0.7897 $\pm$ 0.0033 | 0.05101 $\pm$ 0.0021 | 0.9677 $\pm$ 0.0056 | 0.8555 $\pm$ 0.0032
| LDLSF | 0.1009 $\pm$ 0.0038 | 0.4199 $\pm$ 0.0044 | 0.9008 $\pm$ 0.0015 | 0.05199 $\pm$ 0.0040 | 0.9780 $\pm$ 0.0029 | 0.8660 $\pm$ 0.0022
| LALOT | 0.0989 $\pm$ 0.0019 | 0.6689 $\pm$ 0.0019 | 0.8089 $\pm$ 0.0049 | 0.04778 $\pm$ 0.0018 | 0.9476 $\pm$ 0.0020 | 0.8700 $\pm$ 0.0033
SBU | BFGS-LLD | 0.1119 $\pm$ 0.0030 | 1.4657 $\pm$ 0.0022 | 0.7700 $\pm$ 0.0025 | 0.04932 $\pm$ 0.0053 | 0.9753 $\pm$ 0.0036 | 0.8710 $\pm$ 0.0019
| Ours | 0.2998 $\pm$ 0.0020 | 2.3374 $\pm$ 0.0018 | 6.5163 $\pm$ 0.0018 | 0.8111 $\pm$ 0.0029 | 0.7890 $\pm$ 0.0049 | 0.5691 $\pm$ 0.0010
| LDL-LRR | 0.3889 $\pm$ 0.0111 | 3.1698 $\pm$ 0.0031 | 6.8777 $\pm$ 0.0025 | 0.8999 $\pm$ 0.0069 | 0.7044 $\pm$ 0.0077 | 0.5444 $\pm$ 0.0049
| LDL-LCLR | 0.3740 $\pm$ 0.0066 | 2.4986 $\pm$ 0.0066 | 6.8600 $\pm$ 0.0067 | 0.8559 $\pm$ 0.0039 | 0.7119 $\pm$ 0.0122 | 0.5119 $\pm$ 0.0081
| LDLSF | 0.3441 $\pm$ 0.0249 | 2.9884 $\pm$ 0.0055 | 6.6900 $\pm$ 0.0055 | 0.8391 $\pm$ 0.0044 | 0.7336 $\pm$ 0.0088 | 0.8660 $\pm$ 0.0041
| LALOT | 0.3129 $\pm$ 0.0152 | 2.3999 $\pm$ 0.0044 | 6.6666 $\pm$ 0.0078 | 0.8226 $\pm$ 0.0033 | 0.7390 $\pm$ 0.0100 | 0.5224 $\pm$ 0.0066
Scene | BFGS-LLD | 0.3598 $\pm$ 0.0020 | 2.4998 $\pm$ 0.0033 | 6.7999 $\pm$ 0.0049 | 0.8400 $\pm$ 0.0033 | 0.7333 $\pm$ 0.0064 | 0.5199 $\pm$ 0.0055
| Ours | 0.0488 $\pm$ 0.0012 | 2.1029 $\pm$ 0.0259 | 14.0888 $\pm$ 0.0551 | 0.2335 $\pm$ 0.0044 | 0.8395 $\pm$ 0.0032 | 0.7984 $\pm$ 0.0066
| LDL-LRR | 0.0537 $\pm$ 0.0039 | 2.2887 $\pm$ 0.0860 | 14.3550 $\pm$ 0.0144 | 0.2559 $\pm$ 0.0077 | 0.8288 $\pm$ 0.0144 | 0.7789 $\pm$ 0.0040
| LDL-LCLR | 0.0511 $\pm$ 0.0022 | 2.2201 $\pm$ 0.0444 | 14.2101 $\pm$ 0.0510 | 0.2566 $\pm$ 0.0047 | 0.8302 $\pm$ 0.0012 | 0.7722 $\pm$ 0.0060
| LDLSF | 0.0513 $\pm$ 0.0030 | 2.2221$\pm$ 0.0036 | 14.3667 $\pm$ 0.0265 | 0.2445 $\pm$ 0.0077 | 0.8320 $\pm$ 0.0010 | 0.7701 $\pm$ 0.0026
| LALOT | 0.0505 $\pm$ 0.0033 | 2.1989 $\pm$ 0.0194 | 14.1855 $\pm$ 0.0922 | 0.2443 $\pm$ 0.0088 | 0.8297 $\pm$ 0.0060 | 0.7888 $\pm$ 0.0013
Gene | BFGS-LLD | 0.0578 $\pm$ 0.0066 | 2.3008 $\pm$ 0.0188 | 14.3559 $\pm$ 0.1556 | 0.2480 $\pm$ 0.0015 | 0.8300$\pm$ 0.0049 | 0.7786 $\pm$ 0.0070
| Ours | 0.1089 $\pm$ 0.0018 | 0.5001 $\pm$ 0.0044 | 0.9722 $\pm$ 0.0040 | 0.0977 $\pm$ 0.0008 | 0.9485 $\pm$ 0.0061 | 0.8602 $\pm$ 0.0006
| LDL-LRR | 0.1135 $\pm$ 0.0009 | 0.5244$\pm$ 0.0010 | 1.1551 $\pm$ 0.0061 | 0.1445 $\pm$ 0.0049 | 0.9510 $\pm$ 0.0022 | 0.8772 $\pm$ 0.0007
| LDL-LCLR | 0.1177 $\pm$ 0.0086 | 0.5345 $\pm$ 0.0040 | 1.1533 $\pm$ 0.0111 | 0.1559 $\pm$ 0.0030 | 0.9360 $\pm$ 0.0049 | 0.8222 $\pm$ 0.0011
| LDLSF | 0.1155 $\pm$ 0.0045 | 0.5339 $\pm$ 0.0062 | 1.1152$\pm$ 0.0050 | 0.1540 $\pm$ 0.0041 | 0.9445 $\pm$ 0.0020 | 0.8551 $\pm$ 0.0044
| LALOT | 0.1221 $\pm$ 0.0110 | 0.5440 $\pm$ 0.0033 | 1.111 $\pm$ 0.0040 | 0.1503 $\pm$ 0.0008 | 0.9477 $\pm$ 0.0022 | 0.8559 $\pm$ 0.0002
Movie | BFGS-LLD | 0.1310 $\pm$ 0.0032 | 0.5230 $\pm$ 0.0022 | 1.1170 $\pm$ 0.0024 | 0.1595 $\pm$ 0.0155 | 0.9400 $\pm$ 0.0003 | 0.8491 $\pm$ 0.0018
| Ours | 0.3763 $\pm$ 0.0022 | 1.1560 $\pm$ 0.0102 | 2.0889 $\pm$ 0.0055 | 0.4880 $\pm$ 0.0023 | 0.7998 $\pm$ 0.0022 | 0.6703 $\pm$ 0.0033
| LDL-LRR | 0.3993 $\pm$ 0.0010 | 1.4990 $\pm$ 0.0166 | 2.1884 $\pm$ 0.0034 | 0.5246 $\pm$ 0.0006 | 0.7531 $\pm$ 0.0023 | 0.6334 $\pm$ 0.0077
| LDL-LCLR | 0.4040 $\pm$ 0.0082 | 1.2444 $\pm$ 0.0045 | 2.2000 $\pm$ 0.0009 | 0.4996 $\pm$ 0.0013 | 0.7760 $\pm$ 0.0079 | 0.6555 $\pm$ 0.0012
| LDLSF | 0.4159 $\pm$ 0.0055 | 1.3105 $\pm$ 0.0041 | 2.2155 $\pm$ 0.0076 | 0.5002 $\pm$ 0.0006 | 0.7552 $\pm$ 0.0004 | 0.6234 $\pm$ 0.0033
| LALOT | 0.3881 $\pm$ 0.0099 | 1.4883 $\pm$ 0.0012 | 2.1257 $\pm$ 0.0268 | 0.4990 $\pm$ 0.0008 | 0.7549 $\pm$ 0.0021 | 0.6620 $\pm$ 0.0053
M2B | BFGS-LLD | 0.3811 $\pm$ 0.0044 | 1.3650 $\pm$ 0.0002 | 2.1992 $\pm$ 0.0095 | 0.4995 $\pm$ 0.0005 | 0.7699 $\pm$ 0.0040 | 0.6532 $\pm$ 0.0009
| Ours | 0.3895 $\pm$ 0.0021 | 1.2640 $\pm$ 0.0111 | 2.1995 $\pm$ 0.0095 | 0.4911 $\pm$ 0.0030 | 0.6990 $\pm$ 0.0002 | 0.6004 $\pm$ 0.0001
| LDL-LRR | 0.4159 $\pm$ 0.0010 | 1.6680 $\pm$ 0.0122 | 2.2006 $\pm$ 0.0039 | 0.5388 $\pm$ 0.0006 | 0.6531 $\pm$ 0.0023 | 0.5804 $\pm$ 0.0007
| LDL-LCLR | 0.4240 $\pm$ 0.0042 | 1.3444 $\pm$ 0.0055 | 2.2450$\pm$ 0.0016 | 0.5131 $\pm$ 0.0022 | 0.6261 $\pm$ 0.0005 | 0.5500 $\pm$ 0.0012
| LDLSF | 0.4360 $\pm$ 0.0015 | 1.2185 $\pm$ 0.0022 | 2.2159 $\pm$ 0.0076 | 0.5120 $\pm$ 0.0006 | 0.6261 $\pm$ 0.0004 | 0.5534 $\pm$ 0.0030
| LALOT | 0.3999 $\pm$ 0.0009 | 1.4983 $\pm$ 0.0012 | 2.22007 $\pm$ 0.0158 | 0.4995 $\pm$ 0.0002 | 0.6549 $\pm$ 0.0020 | 0.6411 $\pm$ 0.0044
SCUT | BFGS-LLD | 0.3992 $\pm$ 0.0055 | 1.5656 $\pm$ 0.0163 | 2.2832 $\pm$ 0.0080 | 0.4966 $\pm$ 0.0011 | 0.6491 $\pm$ 0.0040 | 0.6333 $\pm$ 0.0013
| Ours | 0.1251 $\pm$ 0.0002 | 1.1890 $\pm$ 0.0120 | 2.0980 $\pm$ 0.0223 | 0.1053 $\pm$ 0.0009 | 0.9643 $\pm$ 0.0015 | 0.8501 $\pm$ 0.0025
| LDL-LRR | 0.1313 $\pm$ 0.0031 | 1.2519 $\pm$ 0.0038 | 2.1992 $\pm$ 0.0095 | 0.1127 $\pm$ 0.0077 | 0.9533 $\pm$ 0.0021 | 0.8412 $\pm$ 0.0066
| LDL-LCLR | 0.1277 $\pm$ 0.0016 | 1.1969 $\pm$ 0.0039 | 2.1194 $\pm$ 0.0046 | 0.1135 $\pm$ 0.0006 | 0.9588 $\pm$ 0.0044 | 0.8483 $\pm$ 0.0014
| LDLSF | 0.1270 $\pm$ 0.0028 | 1.1909 $\pm$ 0.0164 | 2.1846 $\pm$ 0.0119 | 0.1193 $\pm$ 0.0041 | 0.9609 $\pm$ 0.0019 | 0.8460 $\pm$ 0.0007
| LALOT | 0.1306 $\pm$ 0.0022 | 1.1921 $\pm$ 0.0015 | 2.1111 $\pm$ 0.0171 | 0.1120 $\pm$ 0.0015 | 0.9430 $\pm$ 0.0019 | 0.8400 $\pm$ 0.0004
fbp5500 | BFGS-LLD | 0.1299 $\pm$ 0.0049 | 1.4655 $\pm$ 0.0041 | 2.1675 $\pm$ 0.0024 | 0.1135 $\pm$ 0.0055 | 0.9595 $\pm$ 0.0030 | 0.8419 $\pm$ 0.0018
| Ours | 0.1456 $\pm$ 0.0021 | 1.3651 $\pm$ 0.0441 | 2.6888 $\pm$ 0.0023 | 0.2017 $\pm$ 0.0012 | 0.9394 $\pm$ 0.0026 | 0.8247 $\pm$ 0.0077
| LDL-LRR | 0.1526 $\pm$ 0.0033 | 1.5651 $\pm$ 0.0111 | 2.7594 $\pm$ 0.0422 | 0.2449 $\pm$ 0.0007 | 0.9251 $\pm$ 0.0003 | 0.8141 $\pm$ 0.0044
| LDL-LCLR | 0.1515 $\pm$ 0.0022 | 1.592 $\pm$ 0.0117 | 2.7779 $\pm$ 0.0239 | 0.2244 $\pm$ 0.0030 | 0.9262 $\pm$ 0.0062 | 0.8189 $\pm$ 0.0098
| LDLSF | 0.1488 $\pm$ 0.0024 | 1.3889$\pm$ 0.0086 | 2.7672 $\pm$ 0.0660 | 0.2302 $\pm$ 0.0044 | 0.9111 $\pm$ 0.0051 | 0.8117 $\pm$ 0.0022
| LALOT | 0.1479 $\pm$ 0.0010 | 1.3659 $\pm$ 0.0099 | 2.6956 $\pm$ 0.0144 | 0.2221 $\pm$ 0.0064 | 0.9311 $\pm$ 0.0021 | 0.8107 $\pm$ 0.0008
RAF-ML | BFGS-LLD | 0.1499 $\pm$ 0.0009 | 1.6656$\pm$ 0.0066 | 2.7101 $\pm$ 0.0211 | 0.2541 $\pm$ 0.0055 | 0.9204 $\pm$ 0.0023 | 0.8157 $\pm$ 0.0050
| Ours | 0.2777 $\pm$ 0.0021 | 2.2374 $\pm$ 0.0110 | 5.1163 $\pm$ 0.0018 | 0.5111 $\pm$ 0.0029 | 0.8807 $\pm$ 0.0049 | 0.7891 $\pm$ 0.0014
| LDL-LRR | 0.3129 $\pm$ 0.0021 | 3.2441 $\pm$ 0.0031 | 6.1454 $\pm$ 0.0023 | 0.6616 $\pm$ 0.0035 | 0.8002 $\pm$ 0.0042 | 0.7411 $\pm$ 0.0014
| LDL-LCLR | 0.2994$\pm$ 0.0045 | 2.4900 $\pm$ 0.0012 | 6.9609 $\pm$ 0.0041 | 0.6056 $\pm$ 0.0031 | 0.7110 $\pm$ 0.0021 | 0.7110 $\pm$ 0.0088
| LDLSF | 0.3007 $\pm$ 0.0002 | 2.7887 $\pm$ 0.0057 | 5.6101 $\pm$ 0.0118 | 0.6396 $\pm$ 0.0022 | 0.7939 $\pm$ 0.0098 | 0.7660 $\pm$ 0.0007
| LALOT | 0.3133 $\pm$ 0.0021 | 2.3141 $\pm$ 0.0016 | 5.5336 $\pm$ 0.0241 | 0.5233 $\pm$ 0.0012 | 0.8595 $\pm$ 0.055 | 0.7214 $\pm$ 0.0049
Twitter | BFGS-LLD | 0.3114 $\pm$ 0.0044 | 2.5511 $\pm$ 0.0028 | 5.7145 $\pm$ 0.0041 | 0.5461 $\pm$ 0.0153 | 0.8335 $\pm$ 0.0055 | 0.7744 $\pm$ 0.0020
| Ours | 0.2816 $\pm$ 0.0031 | 2.3356 $\pm$ 0.0097 | 5.2222 $\pm$ 0.0159 | 0.5314 $\pm$ 0.0033 | 0.8406 $\pm$ 0.0041 | 0.7741 $\pm$ 0.0025
| LDL-LRR | 0.3329 $\pm$ 0.0012 | 3.4400 $\pm$ 0.0174 | 6.3459 $\pm$ 0.0229 | 0.6516 $\pm$ 0.0031 | 0.8450$\pm$ 0.0040 | 0.7399 $\pm$ 0.0037
| LDL-LCLR | 0.2970$\pm$ 0.0009 | 2.4444 $\pm$ 0.0063 | 6.1600 $\pm$ 0.0041 | 0.6222 $\pm$ 0.0013 | 0.7919 $\pm$ 0.0029 | 0.7090$\pm$ 0.0070
| LDLSF | 0.3301 $\pm$ 0.0009 | 2.8888 $\pm$ 0.0459 | 5.9152 $\pm$ 0.0121 | 0.6100 $\pm$ 0.0021 | 0.8139 $\pm$ 0.0098 | 0.7360 $\pm$ 0.0037
| LALOT | 0.3411 $\pm$ 0.0026 | 2.9140$\pm$ 0.0019 | 5.3333 $\pm$ 0.0243 | 0.5737 $\pm$ 0.0012 | 0.8225 $\pm$ 0.020 | 0.7144 $\pm$ 0.0004
Flickr | BFGS-LLD | 0.3200$\pm$ 0.0041 | 2.7517 $\pm$ 0.0060 | 5.8149 $\pm$ 0.0048 | 0.5961 $\pm$ 0.0099 | 0.8131 $\pm$ 0.0011 | 0.7407 $\pm$ 0.0077

Results and Discussion.

We conduct 10 times 5-fold cross-validation on each dataset. The experimental
results are presented in the form of “mean$\pm$std” in Table 3. Overall, our
proposed method outperforms other comparison algorithms in all evaluation
metrics. Each comparison algorithm employs some regularization techniques to
expand the training sample as well as to prevent overfitting, however, three
main factors contribute to the competitive results of our approach. i):
Moderate noise, especially on the Gene dataset, due to the uncertainty that
comes with manual annotation, our approach has a huge performance gain with
the help of implicit distribution representation with Gaussian priors. ii):
The ability to capture global features between labels with the help of a self-
attentive mechanism. Also, LALOT performs sub-optimally probably because of
global modeling. iii): The powerful representational capabilities of the deep
network, especially on image datasets, give us a huge advantage.

Table 4: Ablation study (AS). Effectiveness of the loss functions and the
modules on two datasets. Quantitative results demonstrate the effectiveness of
each module.

AS | Algorithm | Chebyshev $\downarrow$ | Clark $\downarrow$ | Canberra $\downarrow$ | K-L $\downarrow$ | Cosine $\uparrow$ | Intersection $\uparrow$ | Dataset
---|---|---|---|---|---|---|---|---
| Ours | 0.0488 $\pm$ 0.0012 | 2.1029 $\pm$ 0.0259 | 14.0888 $\pm$ 0.0551 | 0.2335 $\pm$ 0.0044 | 0.8395 $\pm$ 0.0032 | 0.7984 $\pm$ 0.0066 |
(a) | w/o $\mathcal{L}_{p}$ | 0.0497 $\pm$ 0.0021 | 2.1664 $\pm$ 0.0177 | 14.2651 $\pm$ 0.0155 | 0.2488 $\pm$ 0.0071 | 0.8290 $\pm$ 0.074 | 0.7884 $\pm$ 0.0037 | Gene
| Ours | 0.0779 $\pm$ 0.0021 | 0.3980 $\pm$ 0.0051 | 0.7779 $\pm$ 0.0030 | 0.04040 $\pm$ 0.0020 | 0.9883 $\pm$ 0.0009 | 0.8778 $\pm$ 0.0014 |
| w/o PRT | 0.0877 $\pm$ 0.0009 | 0.4008 $\pm$ 0.0043 | 0.7881 $\pm$ 0.0014 | 0.04223 $\pm$ 0.0010 | 0.9779 $\pm$ 0.0008 | 0.8699 $\pm$ 0.0012 |
| SNN | 0.0771 $\pm$ 0.0012 | 0.4006 $\pm$ 0.0047 | 0.7805 $\pm$ 0.0011 | 0.04118 $\pm$ 0.0016 | 0.9801 $\pm$ 0.0012 | 0.8664 $\pm$ 0.0029 |
(b,c,d) | GNN | 0.0804 $\pm$ 0.0033 | 0.4133 $\pm$ 0.0017 | 0.7968 $\pm$ 0.0020 | 0.04991 $\pm$ 0.0052 | 0.9705 $\pm$ 0.0036 | 0.8612 $\pm$ 0.0016 | wc-LDL

Ablation Study.

To demonstrate the effectiveness of the loss function and the module of our
model, we conduct an ablation study involving the following four experiments:
(a) w/o perceptual loss function $\mathcal{L}_{p}$: we remove the loss
function on training Gene dataset, shown in Table 4. (b) w/o our proposed
regularization techniques (PRT): we remove the linear normalization function
and the data augmentation respectively on the training wc-LDL dataset, shown
in Table 4. (c) The effectiveness of SNN: we use standard MLPs to replace SNNs
in the same network architecture, shown in Table 4. (d) The effectiveness of
GNN: for deep implicit function construction, we use standard MLPs to replace
GNNs, as shown in Table 4. We conduct 10 times 5-fold cross-validation on the
dataset of the ablation experiment.

Potential of Model and Network.

Our model can be extended to handle classification and semi-supervised tasks.
We conduct some experiment reports to show the potential of the public
datasets. In addition, there is a deep model (DLDL (Gao et al. (2017))) based
on a label distribution learning framework as a training scheme being
performed on several classification tasks. Since DLDL does not have an adapted
network implementation on tabular data, to demonstrate the effectiveness of
our proposed method, our approach battles with the DLDL algorithm in the age
estimation dataset.

(1) Facial expression recognition (Zhao et al. (2021)): To evaluate the
effectiveness of the proposed model, we conduct the experiments on the public
in-the-wild facial expression datasets (RAF-DB, CAER-S, AffectNet). For images
on all the datasets, the face region is detected and aligned by using
Retinaface (Deng et al. (2020)). Then, the image is cropped to a fixed
resolution (224 $\times$ 224) by bilinear interpolation. Our approach is pre-
trained on the face recognition dataset MS-Celeb-1M, and the 50-layer Residual
Network is used as the backbone network. For our network, parameters were
optimized using the Adam optimizer with an initial learning rate of 0.01, a
mini-batch size of 128, and an epoch is 50. Notably, our network architecture
uses a model (both input and output layers are changed in the number of
neurons and the rest are fixed) executed on the Gene dataset with a single RTX
GPU, and the loss function uses cross-entropy. We compared it with the current
SOTA model (Face2Exp) on three popular datasets. The recognition accuracy of
Face2Exp in the three data sets (RAF-DB, CAER-S, AffectNet) is 88.54%, 86.16%,
and 64.23% respectively. The recognition accuracy of our proposed model in
three datasets is 88.52%, 86.36%, and 65.02% respectively. Our algorithm
achieves competitive results on most of the face recognition datasets. (2)
MedMNIST Classification Decathlon (Yang et al. (2021)): We evaluate the
algorithm’s performance on the MedMNIST Classification Decathlon benchmark.
The area under the ROC curve (AUC) and Accuracy (ACC) is used as the
evaluation metrics. Our approach is pre-trained on the Gene dataset, the
18-layer Residual Network is adopted as the backbone network. Our model is
trained for 100 epochs, using a cross-entropy loss and an Adam optimizer with
a batch size of 128 and an initial learning rate of $1\times 10^{-3}$ The
overall performance of the methods is reported in Table 5. Our model is pre-
trained during the training phase and thus achieves competitive performance on
most of the datasets. Literature (Yang et al. (2021)) includes the
bibliography of the full comparison method.

Table 5: Overall performance of MedMNIST in metrics of AUC and ACC, using ResNet-18 / ResNet-50 with resolution $28$ and $224$, auto-sklearn , AutoKeras and Google AutoML Vision. Methods | PathMNIST | ChestMNIST | DermaMNIST | OCTMNIST | PneumoniaMNIST
---|---|---|---|---|---
AUC | ACC | AUC | ACC | AUC | ACC | AUC | ACC | AUC | ACC
ResNet-18 (28) | 0.972 | 0.844 | 0.706 | 0.947 | 0.899 | 0.721 | 0.951 | 0.758 | 0.957 | 0.843
ResNet-18 (224) | 0.978 | 0.860 | 0.713 | 0.948 | 0.896 | 0.727 | 0.960 | 0.752 | 0.970 | 0.861
ResNet-50 (28) | 0.979 | 0.864 | 0.692 | 0.947 | 0.886 | 0.710 | 0.939 | 0.745 | 0.949 | 0.857
ResNet-50 (224) | 0.978 | 0.848 | 0.706 | 0.947 | 0.895 | 0.719 | 0.951 | 0.750 | 0.968 | 0.896
auto-sklearn | 0.500 | 0.186 | 0.647 | 0.642 | 0.906 | 0.734 | 0.883 | 0.595 | 0.947 | 0.865
AutoKeras | 0.979 | 0.864 | 0.715 | 0.939 | 0.921 | 0.756 | 0.956 | 0.736 | 0.970 | 0.918
Google AutoML Vision | 0.982 | 0.811 | 0.718 | 0.947 | 0.925 | 0.766 | 0.965 | 0.732 | 0.993 | 0.941
Ours | 0.985 | 0.864 | 0.722 | 0.948 | 0.929 | 0.765 | 0.969 | 0.756 | 0.992 | 0.943
Methods | RetinaMNIST | BreastMNIST | OrganMNIST_A | OrganMNIST_C | OrganMNIST_S
AUC | ACC | AUC | ACC | AUC | ACC | AUC | ACC | AUC | ACC
ResNet-18 (28) | 0.727 | 0.515 | 0.897 | 0.859 | 0.995 | 0.921 | 0.990 | 0.889 | 0.967 | 0.762
ResNet-18 (224) | 0.721 | 0.543 | 0.915 | 0.878 | 0.997 | 0.931 | 0.991 | 0.907 | 0.974 | 0.777
ResNet-50 (28) | 0.719 | 0.490 | 0.879 | 0.853 | 0.995 | 0.916 | 0.990 | 0.893 | 0.968 | 0.746
ResNet-50 (224) | 0.717 | 0.555 | 0.863 | 0.833 | 0.997 | 0.931 | 0.992 | 0.898 | 0.970 | 0.770
auto-sklearn | 0.694 | 0.525 | 0.848 | 0.808 | 0.797 | 0.563 | 0.898 | 0.676 | 0.855 | 0.601
AutoKeras | 0.655 | 0.420 | 0.833 | 0.801 | 0.996 | 0.929 | 0.992 | 0.915 | 0.972 | 0.803
Google AutoML Vision | 0.762 | 0.530 | 0.932 | 0.865 | 0.988 | 0.818 | 0.986 | 0.861 | 0.964 | 0.706
Ours | 0.760 | 0.570 | 0.931 | 0.890 | 0.996 | 0.933 | 0.990 | 0.917 | 0.971 | 0.810

(3) CIFAR-10, SVHN, CIFAR-100 (Hu et al. (2021)): Since our approach involves
graph structure, it battles with other SOTAs on the semi-supervised task. We
use ResNet-28-2 as our backbone and Adam with weight decay for optimization in
all experiments. Our model is trained for 100 epochs, using a cross-entropy
loss with a batch size of 128 and an initial learning rate of $1\times
10^{-2}$ Early-stopping and data augmentation (Mixup) are also adopted in the
training phase. Two datasets (CIFAR-10, and SVHN) are used for us to evaluate
the performance of the algorithm. The overall performance of the methods is
reported in Table 6. Our model still selects MLPs pre-trained on the Gene
dataset and performs competitively on the semi-supervised task. Literature (Hu
et al. (2021)) includes the bibliography of the full comparison method.

Table 6: CIFAR-10, CIFAR-100, and SVHN Top-1 test accuracy. Methods | CIFAR-10 | SVHN | CIFAR-100 | Backbone
---|---|---|---|---
1000 labels | 4000 labels | 1000 labels | 4000 labels | 10000 labels | 15000 labels
VAT | 81.36 | 88.95 | 94.02 | 95.80 | 77.54 | 81.55 | WRN-28-8
MeanTeacher | 82.68 | 89.64 | 96.25 | 96.61 | 72.68 | 79.62 | WRN-28-8
MixMatch | 92.25 | 93.76 | 96.73 | 97.11 | 71.69 | 79.13 | WRN-28-8
ReMixMatch | 94.27 | 94.86 | 97.17 | 97.58 | 76.97 | 80.99 | WRN-28-8
FixMatch | - | 95.69 | 97.64 | - | 77.40 | 64.25 | WRN-28-8
SimPLE | 94.84 | 94.95 | 97.54 | 97.31 | 78.11 | 82.40 | WRN-28-8
Ours | 94.98 | 95.99 | 97.56 | 97.62 | 78.64 | 81.02 | WRN-28-8

(4) Semi-supervised label distribution learning (Jia et al. (2021b)): To
evaluate the capability of our model, our proposed algorithm is compared to
the SOTA model (PGE-SLDL (Jia et al. (2021b))) in the Gene with a 50% missing
rate. We conduct it 10 times on Gene and average the 10 results as the final
result. PGE-SLDL yield 0.0059$\pm$0.0008, 0.4298$\pm$0.0006,
4.3899$\pm$0.0005, 0.09788$\pm$0.0005, 0.8911$\pm$0.0008 and 0.8789$\pm$0.0007
respectively on the data set with 6 evaluation metrics (Chebyshev$\downarrow$,
Clark$\downarrow$, Canberra$\downarrow$, K-L divergence$\downarrow$,
Intersection$\uparrow$, Cosine$\uparrow$). Our method yield 0.0053$\pm$0.0012,
0.4112$\pm$0.0005, 4.2990$\pm$0.0008, 0.09661$\pm$0.0005, 0.9001$\pm$0.0012
and 0.8797$\pm$0.0008 respectively on the data set with corresponding 6
evaluation metrics. Our method still shows competitiveness, where the model
remains fixed, the learning rate and the number of iterations are not changed,
and the $\lambda_{2}$ of Equation 6 is adjusted to 0.25. (5) Our approach vs.
DLDL (Age estimation (Gao et al. (2017))): Two age estimation datasets are
used in our experiments. The first is Morph (Ricanek & Tesafaye (2006)), which
is one of the largest publicly available age datasets. The second dataset is
from the apparent age estimation competition, the first competition track of
the ICCV ChaLearn LAP 2015 workshop (Escalera et al. (2015)). We employ the
DPM model to detect the main facial region. Then, the detected face is fed
into a cascaded convolution network to get the five facial key points,
including the left/right eye centers, nose tip, and left/right mouth corners.
Finally, based on these facial points, we align the face to the upright pose.
The implementation details of our method with the preprocessing unit are
referenced in DLDL. We used MAE and $\epsilon$-error to evaluate the
performance of our model and the comparison method, respectively. DLDL
algorithm on Morph and ChaLearn with metrics MAE $\downarrow$ and
$\epsilon$-error yields performance that is {2.42, 0.23; 3.51, 0.31}. Our
algorithm achieves competitive results ({2.40, 0.20; 3.23, 0.27}).

Visualize of Label Distribution Matrix.

We conduct an experiment to verify the validity of our algorithm. We extract
one sample (including features and labeled distributions) on Movie, and then
visualize the matrix of label distribution ($5\times 10$) learned by the
network and the corresponding label distribution. As shown in the Figure 2, we
notice that the data distribution (means and variances) of the label
distribution vector (each row of the label distribution matrix) is consistent
with the corresponding label distribution.

|  |
---|---|---
(a) L1 of the matrix | (b) L2 of the matrix | (c) L3 of the matrix
|  |
(d) L4 of the matrix | (e) L5 of the matrix | (f) Label distribution
Figure 2: This figure shows the distribution of data for each vector of the
label distribution matrix and the corresponding label for this label
distribution matrix.

##  4 Application and Energy Consumption

We select the algorithm executed on Gene as the evaluation model. The
evaluation model is conducted on three platforms (lynxi HP300, Raspberry Pi,
and Apple smartphones) to check the energy consumption with the same number of
iterations (the power is evaluated thanks to the adb script (Dzhagaryan et al.
(2016))). Specifically, our model is trained on Gene with the accuracy of
float16. Then, the baseline model (MLP replaces SNN and GNN) and our model run
inference on a mobile platform, each model executing 500 epochs with 32 batch
sizes in the iteration. Experimental results show that our algorithm saves
34.6%, 41.2%, and 40.8% energy compared to the baseline algorithm on the three
platforms, respectively.

##  5 Related Works

Label distribution Learning.

Label distribution learning has attracted several attention as a new learning
paradigm. Label distribution learning comes from the scheme proposed by (Geng
(2016)) to address the age estimation task. Since then a large number of
approaches have been proposed, such as low-rank hypothesis-based (Jia et al.
(2019); Ren et al. (2019b)), metric-based (Gao et al. (2018)), manifold-based
(Wang & Geng (2021)), and label correlation-based (Teng & Jia (2021); Qian et
al. (2022)). Among them, some approaches are executed in computer vision (Chen
et al. (2021a)), and speech recognition (Si et al. (2022)) tasks to improve
the performance of classifiers. In this paper, we try to build a distribution
of label distributions to moderate noise and uncertainty.

Implicit Neural Representations. In implicit neural representation, an object
is usually represented as a multi-layer perception (MLP) that maps coordinates
to a signal. This idea has been widely applied in modeling 3D object shapes
(Lin et al. (2020); Kohli et al. (2020)), 3D surfaces of the scene (Sitzmann
et al. (2019); Yariv et al. (2020); Niemeyer et al. (2020); Jiang et al.
(2020)), the appearance of the 3D structure as well as the 2D image
enhancement (Skorokhodov et al. (2021); Chen et al. (2021b); Anokhin et al.
(2021); Karras et al. (2021)). In this paper, we seek to explore this
technique to address the label distribution learning issue.

##  6 Conclusion

In this paper, we design an implicit distribution representation algorithm to
moderate the uncertainty of the label values, where the implicit function can
be a good estimate of the continuous distribution space. Furthermore, Gaussian
prior methods and self-attention mechanisms help the model learn both local
signals and global information of the label distribution matrix. Numerous
experiments have verified the effectiveness of our approach as well as the
suitability of the regularization technique. The application session
demonstrates the high efficiency of our proposed model.

## References

  * Anokhin et al. (2021) Ivan Anokhin, Kirill Demochkin, Taras Khakhulin, Gleb Sterkin, Victor Lempitsky, and Denis Korzhenkov.  Image generators with conditionally-independent pixel synthesis.  In _CVPR_ , 2021.
  * Chen et al. (2021a) Jingying Chen, Chen Guo, Ruyi Xu, Kun Zhang, Zongkai Yang, and Honghai Liu.  Toward children’s empathy ability analysis: Joint facial expression recognition and intensity estimation using label distribution learning.  _IEEE Transactions on Industrial Informatics_ , 2021a.
  * Chen et al. (2021b) Yinbo Chen, Sifei Liu, and Xiaolong Wang.  Learning continuous image representation with local implicit image function.  In _CVPR_ , 2021b.
  * Deng et al. (2020) Jiankang Deng, Jia Guo, Evangelos Ververas, Irene Kotsia, and Stefanos Zafeiriou.  Retinaface: Single-shot multi-level face localisation in the wild.  In _CVPR_ , 2020.
  * Dzhagaryan et al. (2016) Armen Dzhagaryan, Aleksandar Milenkovic, Mladen Milosevic, and Emil Jovanov.  An environment for automated measuring of energy consumed by android mobile devices.  In _PECCS_ , 2016.
  * Escalera et al. (2015) Sergio Escalera, Junior Fabian, Pablo Pardo, Xavier Baró, Jordi Gonzalez, Hugo J Escalante, Dusan Misevic, Ulrich Steiner, and Isabelle Guyon.  Chalearn looking at people 2015: Apparent age and cultural event recognition datasets and results.  In _ICCVW_ , 2015.
  * Gao et al. (2017) Bin-Bin Gao, Chao Xing, Chen-Wei Xie, Jianxin Wu, and Xin Geng.  Deep label distribution learning with label ambiguity.  _IEEE TIP_ , 2017.
  * Gao et al. (2018) Bin-Bin Gao, Hong-Yu Zhou, Jianxin Wu, and Xin Geng.  Age estimation using expectation of label distribution learning.  In _IJCAI_ , 2018.
  * Geng (2016) Xin Geng.  Label distribution learning.  _IEEE TKDE_ , 2016.
  * Ghosh et al. (2021) Aishik Ghosh, Benjamin Nachman, and Daniel Whiteson.  Uncertainty-aware machine learning for high energy physics.  _Physical Review D_ , 2021.
  * He et al. (2015) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Delving deep into rectifiers: Surpassing human-level performance on imagenet classification.  In _ICCV_ , 2015.
  * Hu et al. (2021) Zijian Hu, Zhengyu Yang, Xuefeng Hu, and Ram Nevatia.  Simple: similar pseudo label exploitation for semi-supervised classification.  In _CVPR_ , 2021.
  * Jia et al. (2019) Xiuyi Jia, Xiang Zheng, Weiwei Li, Changqing Zhang, and Zechao Li.  Facial emotion distribution learning by exploiting low-rank label correlations locally.  In _CVPR_ , 2019.
  * Jia et al. (2021a) Xiuyi Jia, Xiaoxia Shen, Weiwei Li, Yunan Lu, and Jihua Zhu.  Label distribution learning by maintaining label ranking relation.  _IEEE TKDE_ , 2021a.
  * Jia et al. (2021b) Xiuyi Jia, Tao Wen, Weiping Ding, Huaxiong Li, and Weiwei Li.  Semi-supervised label distribution learning via projection graph embedding.  _Information Sciences_ , 2021b.
  * Jiang et al. (2020) Chiyu Jiang, Avneesh Sud, Ameesh Makadia, Jingwei Huang, Matthias Nießner, Thomas Funkhouser, et al.  Local implicit grid representations for 3D scenes.  In _CVPR_ , 2020.
  * Johnson et al. (2016) Justin Johnson, Alexandre Alahi, and Li Fei-Fei.  Perceptual losses for real-time style transfer and super-resolution.  In _ECCV_ , 2016.
  * Ju et al. (2019) Mingye Ju, Can Ding, Y Jay Guo, and Dengyin Zhang.  Idgcp: Image dehazing based on gamma correction prior.  _IEEE TIP_ , 2019.
  * Karras et al. (2021) Tero Karras, Miika Aittala, Samuli Laine, Erik Härkönen, Janne Hellsten, Jaakko Lehtinen, and Timo Aila.  Alias-free generative adversarial networks.  In _NeurIPS_ , 2021.
  * Kohli et al. (2020) Amit Kohli, Vincent Sitzmann, and Gordon Wetzstein.  Inferring semantic information with 3D neural scene representations.  _arXiv preprint arXiv:2003.12673_ , 2020.
  * Li et al. (2022) Qiang Li, Jingjing Wang, Zhaoliang Yao, Yachun Li, Pengju Yang, Jingwei Yan, Chunmao Wang, and Shiliang Pu.  Unimodal-concentrated loss: Fully adaptive label distribution learning for ordinal regression.  In _CVPR_ , 2022.
  * Lin et al. (2020) Chen-Hsuan Lin, Chaoyang Wang, and Simon Lucey.  SDF-SRN: Learning signed distance 3D object reconstruction from static images.  In _NeurIPS_ , 2020.
  * Liu et al. (2021) Tingting Liu, Jixin Wang, Bing Yang, and Xuan Wang.  Ngdnet: Nonuniform gaussian-label distribution learning for infrared head pose estimation and on-task behavior understanding in the classroom.  _Neurocomputing_ , 2021.
  * Niemeyer et al. (2020) Michael Niemeyer, Lars Mescheder, Michael Oechsle, and Andreas Geiger.  Differentiable volumetric rendering: Learning implicit 3D representations without 3D supervision.  In _CVPR_ , 2020.
  * Northcutt et al. (2021) Curtis Northcutt, Lu Jiang, and Isaac Chuang.  Confident learning: Estimating uncertainty in dataset labels.  _Journal of Artificial Intelligence Research_ , 2021.
  * Patel et al. (2021) Kinjal Patel, Eric Hunsberger, Sean Batir, and Chris Eliasmith.  A spiking neural network for image segmentation.  _arXiv preprint arXiv:2106.08921_ , 2021.
  * Qian et al. (2022) Wenbin Qian, Yinsong Xiong, Jun Yang, and Wenhao Shu.  Feature selection for label distribution learning via feature similarity and label correlation.  _Information Sciences_ , 2022.
  * Qin et al. (2022) Zhen Qin, Weixuan Sun, Hui Deng, Dongxu Li, Yunshen Wei, Baohong Lv, Junjie Yan, Lingpeng Kong, and Yiran Zhong.  cosformer: Rethinking softmax in attention.  In _ICLR_ , 2022.
  * Ren et al. (2019a) Tingting Ren, Xiuyi Jia, Weiwei Li, Lei Chen, and Zechao Li.  Label distribution learning with label-specific features.  In _IJCAI_ , 2019a.
  * Ren et al. (2019b) Tingting Ren, Xiuyi Jia, Weiwei Li, and Shu Zhao.  Label distribution learning with label correlations via low-rank approximation.  In _IJCAI_ , 2019b.
  * Ricanek & Tesafaye (2006) Karl Ricanek and Tamirat Tesafaye.  Morph: A longitudinal image database of normal adult age-progression.  In _FGR06_ , 2006.
  * Si et al. (2022) Shijing Si, Jianzong Wang, Junqing Peng, and Jing Xiao.  Towards speaker age estimation with label distribution learning.  In _ICASSP_ , 2022.
  * Sitzmann et al. (2019) Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein.  Scene representation networks: Continuous 3D-structure-aware neural scene representations.  In _NeurIPS_ , 2019.
  * Sitzmann et al. (2020) Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein.  Implicit neural representations with periodic activation functions.  In _NeurIPS_ , 2020.
  * Skorokhodov et al. (2021) Ivan Skorokhodov, Savva Ignatyev, and Mohamed Elhoseiny.  Adversarial generation of continuous images.  In _CVPR_ , 2021.
  * Teng & Jia (2021) Qifa Teng and Xiuyi Jia.  Incomplete label distribution learning by exploiting global sample correlation.  In _Multimedia Understanding with Less Labeling on Multimedia Understanding with Less Labeling_. 2021.
  * Ucar et al. (2021) Talip Ucar, Ehsan Hajiramezanali, and Lindsay Edwards.  Subtab: Subsetting features of tabular data for self-supervised representation learning.  In _NeurIPS_ , 2021.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  In _NeurIPS_ , 2017.
  * Wang & Geng (2021) Jing Wang and Xin Geng.  Label distribution learning by exploiting label distribution manifold.  _IEEE TNNLS_ , 2021.
  * Wortsman et al. (2022) Mitchell Wortsman, Gabriel Ilharco, Samir Ya Gadre, Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S Morcos, Hongseok Namkoong, Ali Farhadi, Yair Carmon, Simon Kornblith, et al.  Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time.  In _ICML_ , 2022.
  * Xu et al. (2019) Ning Xu, Yun-Peng Liu, and Xin Geng.  Label enhancement for label distribution learning.  _IEEE TKDE_ , 2019.
  * Xu et al. (2020) Ning Xu, Jun Shu, Yun-Peng Liu, and Xin Geng.  Variational label enhancement.  In _ICML_ , 2020.
  * Yamazaki et al. (2022) Kashu Yamazaki, Viet-Khoa Vo-Ho, Darshan Bulsara, and Ngan Le.  Spiking neural networks and their applications: A review.  _Brain Sciences_ , 2022.
  * Yang et al. (2021) Jiancheng Yang, Rui Shi, and Bingbing Ni.  Medmnist classification decathlon: A lightweight automl benchmark for medical image analysis.  In _ISBI_ , 2021.
  * Yariv et al. (2020) Lior Yariv, Yoni Kasten, Dror Moran, Meirav Galun, Matan Atzmon, Basri Ronen, and Yaron Lipman.  Multiview neural surface reconstruction by disentangling geometry and appearance.  2020\.
  * Zhang et al. (2018) Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz.  Mixup: Beyond empirical risk minimization.  In _ICLR_ , 2018.
  * Zhao & Zhou (2018) Peng Zhao and Zhi-Hua Zhou.  Label distribution learning by optimal transport.  In _AAAI_ , 2018.
  * Zhao et al. (2022) Xingyu Zhao, Yuexuan An, Ning Xu, and Xin Geng.  Fusion label enhancement for multi-label learning.  In _IJCAI_ , 2022.
  * Zhao et al. (2021) Zengqun Zhao, Qingshan Liu, and Feng Zhou.  Robust lightweight facial expression recognition network with label distribution training.  In _AAAI_ , 2021.
  * Zheng et al. (2021a) Qinghai Zheng, Jihua Zhu, Haoyu Tang, Xinyuan Liu, Zhongyu Li, and Huimin Lu.  Generalized label enhancement with sample correlations.  _IEEE TKDE_ , 2021a.
  * Zheng et al. (2021b) Rui Zheng, Shulin Zhang, Lei Liu, Yuhao Luo, and Mingzhai Sun.  Uncertainty in bayesian deep label distribution learning.  _Applied Soft Computing_ , 2021b.


# Coherent pumping of high momentum magnons by light

Fran Šimić  Kavli Institute of NanoScience, Delft University of Technology,
2628 CJ Delft, The Netherlands  Sanchar Sharma  Kavli Institute of
NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands
Yaroslav M. Blanter  Kavli Institute of NanoScience, Delft University of
Technology, 2628 CJ Delft, The Netherlands  Gerrit E. W. Bauer  Institute for
Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577,
Japan  Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ
Delft, The Netherlands

###### Abstract

We propose to excite a large number of coherent magnons with high momentum in
optical cavities. This is achieved by two counterpropagating optical modes
that are detuned by the frequency of a selected magnon, similar to stimulated
Raman scattering. In sub-mm size yttrium iron garnet spheres, a mW laser input
power generates $10^{6}-10^{8}$ coherent magnons. The large magnon population
enhances Brillouin light scattering, a probe suitable to access their quantum
properties.

Magnets are crucial for fast, non-volatile, and robust data storage as well as
candidate materials for logic devices and interconnects [1]. Magnetic
insulators, such as yttrium iron garnet (YIG) [2], are interesting since they
can transport information over long distances via spin waves quantized into
magnons [3; 4], without the Ohmic dissipation of spin transport in metals. The
magnons couple to microwaves [5; 6; 7], electric currents [3; 1; 8],
mechanical motion [9; 10; 11; 12], and light [13; 14]. The high crystal
quality of YIG promises long coherence times [6; 7], opening prospects for
‘quantum magnonics’ [15], the field that strives to employ magnons to store,
process, and transfer information in a quantum coherent manner. Photons can
become a coherent interface to manipulate and probe these magnons.

The GHz magnons in ferro(ferri)magnets interact with light by inelastic
(Brillouin) light scattering (BLS) [13]. By selecting the wave vector of the
input and output photons, e.g. by an optical cavity, specific magnons modes
can be excited [16]. The interaction can be large enough [17; 18] to cool [19]
or herald (generating single magnon states) [20] them, making BLS a promising
probe into their quantum nature. Present experiment focus on the long
wavelength ‘Walker’ (including the ‘Kittel’) magnons in optical resonators
[21; 22; 23; 24; 25]. These have a small overlap with the light fields and
corresponding low intrinsic scattering efficiency, but become observable
because a large magnon density can be resonantly excited by microwaves. On the
other hand, magnons with wavelengths $\sim 100-500\,$nm in the dipolar-
exchange regime have almost perfect overlap with the photon modes in magnetic
spheres [18], but couple only very weakly to microwaves (as do the relevant
magnons in magnetic vortices [17]).

Here, we propose to coherently pump a large number ($\sim 10^{6}-10^{8}$) of
high-momentum magnons by optical lasers, similar to the resonant excitation of
Kittel magnons by microwaves. We exploit the torques exerted by light on the
magnetization by the inverse Faraday and Cotton-Mouton effects [26], which are
proportional to the intensity of the electric field component [26] or, more
precisely, the product of the photon numbers at the incident and scattered
frequencies. Exposing the sample to two phase-coherent lasers that differ in
frequency by a magnon excitation strongly enhances Brillouin scattering [27].
Here we develop the theory of stimulated light scattering by magnons in
optical resonators such as sketched in Fig. 1. Two counter-propagating lasers
feed whispering gallery modes (WGMs) of a YIG sphere via a proximity coupler
such as a fiber or a prism [28; 21; 22; 23]. The WGMs are separated spectrally
by $\sim 1-10\,$GHz, which can be easily tuned into resonance with a magnon by
an applied magnetic field. The two populated WGMs form a spatially-periodic
torque field that excites magnons with matching wavelength. While we focus
here on spherical magnets, the formalism is valid for any magneto-optical
cavity, including planar [29; 30] and cylindrical [17] geometries.

Figure 1: A (massive) sphere of a magnetic insulator, such as YIG, with a
proximity optical coupler, such as a fiber or a prism. Two oppositely
propagating laser beams excite two whispering gallery modes with decay rates
$\kappa_{r,b}$. The photon-magnon scattering coherently amplifies the magnon
amplitude competing with the thermalization rate $\kappa_{m}$.

We consider a minimal model of two WGM modes $\\{W_{r},W_{b}\\}$ resonantly
interacting with a single magnon mode $M$ [see Fig. 1]. We first formulate
heuristic rate equations for the magnon number, $n_{m}^{\mathrm{(sc)}}$ (‘sc’
stands for semi-classical), followed by a more rigorous quantum Langevin
treatment. In the steady state, the energy balance of the processes in Fig. 1
leads to the photon number in the blue sideband $W_{b}$ with frequency
$\omega_{b}$ [31]

$N_{b}=\frac{4K_{b}}{(\kappa_{b}+K_{b})^{2}}\frac{P_{b}}{\hbar\omega_{b}},$
(1)

which is governed by the input light power $P_{b}$, the decay rate
$\kappa_{b}$ in the isolated sphere, and the leakage rate $K_{b}$ into the
proximity coupler. An analogous expression holds for the photon number $N_{r}$
in the red sideband $W_{r}$. Since optomagnonic couplings are small, we
disregarded the backaction exerted by magnons on photons. The reaction rate
for anti-Stokes scattering $W_{r}+M\rightarrow W_{b}$ is
$R_{b}=R_{b}^{(0)}n_{m}^{\mathrm{(sc)}}N_{r}(N_{b}+1),$ while for the reverse
(Stokes) scattering
$R_{r}=R_{r}^{(0)}(n_{m}^{\mathrm{(sc)}}+1)(N_{r}+1)N_{b}$. According to the
Fermi’s golden rule,
$R_{b,r}^{(0)}=2\pi\left|g\right|^{2}\Lambda_{b,r}(\Delta)$, where $g$ is the
matrix element of the Hamiltonian between initial and final states (see
below), the detuning $\Delta\equiv\omega_{b}-\omega_{r}-\omega_{m}$, and

$\Lambda_{b,r}=\frac{1}{2\pi}\frac{\left(\kappa_{b,r}+K_{b,r}\right)}{\Delta^{2}+(\kappa_{b,r}+K_{b,r})^{2}/4},$
(2)

with $\left(\kappa_{b,r}+K_{b,r}\right)^{-1}$ as the photon’s lifetime.

Magnons are lost at a rate
$R_{\mathrm{eq}}=\kappa_{m}\left(n_{m}^{\mathrm{(sc)}}-n_{\mathrm{eq}}\right)$
where $\kappa_{m}^{-1}$ is the magnon lifetime and $n_{\mathrm{eq}}$ is the
equilibrium (Planck) distribution

$n_{\mathrm{eq}}=\left[\exp\left(\frac{\hbar\omega_{m}}{k_{B}T}\right)-1\right]^{-1}.$
(3)

In the steady state $R_{b}+R_{\mathrm{eq}}=R_{r}$

$n_{m}^{\mathrm{(sc)}}=\frac{R_{r}^{(0)}N_{b}\left(N_{r}+1\right)+\kappa_{m}n_{\mathrm{eq}}}{\kappa_{m}+R_{b}^{(0)}N_{r}\left(N_{b}+1\right)-R_{r}^{(0)}\left(N_{r}+1\right)N_{b}}.$
(4)

Eq. (4) agrees with the more rigorous result below only when
$R_{b}^{(0)}=R_{r}^{(0)}$, because here we ignored the correlation between the
forward and backward reactions. Furthermore, the above treatment does not
distinguish between coherent and thermal magnons. For sufficiently large
$N_{r,b}$, $n_{m}^{\mathrm{(sc)}}$ may diverge which is an artifact of
ignoring magnon non-linearities, but such large drives are unrealistic (shown
below).

We consider the 3-particle Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{H}_{\mathrm{om}}$ with non-interacting part

$\hat{H}_{0}=\hbar\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}+\hbar\omega_{b}\hat{a}_{b}^{\dagger}\hat{a}_{b}+\hbar\omega_{m}\hat{m}^{\dagger}\hat{m},$
(5)

where $\\{\hat{a}_{r},\hat{a}_{b},\hat{m}\\}$ are the annihilation operators
for $\\{W_{r},W_{b},M\\},$ respectively. To leading order in the magnon
operators the optomagnonic Hamiltonian is [16]

$\hat{H}_{\mathrm{om}}=\hbar g\hat{a}_{r}\hat{a}_{b}^{\dagger}\hat{m}+\hbar
g^{\ast}\hat{a}_{r}^{\dagger}\hat{a}_{b}\hat{m}^{\dagger}.$ (6)

In the Heisenberg picture, the statistical averages
$\left\langle\hat{X}(t)\right\rangle=\mathrm{Tr}\left[\hat{X}(t)\hat{\rho}\right]$,
where the density matrix $\hat{\rho}=\hat{\rho}_{0}$ is a direct product of an
arbitrary state of the sphere (magnons and WGMs) and a coherent photon state
of the laser input.

The equation of motion for the blue sideband envelope operator
$\hat{W}_{b}\overset{\triangle}{=}\hat{a}_{b}e^{i\omega_{b}t}$ reads [32; 19;
20]

$\frac{d\hat{W}_{b}}{dt}=-ig\hat{W}_{r}\hat{M}e^{i\Delta
t}-\frac{\kappa_{b}+K_{b}}{2}\hat{W}_{b}-\sqrt{\kappa_{b}}\hat{b}_{b}+\sqrt{K_{b}}\hat{A}_{b},$
(7)

where the first term on the r.h.s. is the optomagnonic scattering generated by
the commutator $\left[\hat{a}_{b},\hat{H}\right]$ in the Heisenberg equation.
The second term is the decay of photons inside the sphere,
$\propto\kappa_{b}$, and into the coupler, $\propto K_{b}$. $\hat{b}_{b}$ is
the annihilation field operator of a bath mode that interacts with $W_{b}$
satisfying the commutation relations
$\left[\hat{b}_{b}(t),\hat{b}_{b}^{\dagger}(t^{\prime})\right]=\delta(t-t^{\prime})$
and averages
$\left\langle\hat{b}_{b}(t)\right\rangle=\left\langle\hat{b}_{b}^{\dagger}(t^{\prime})\hat{b}_{b}(t)\right\rangle=0$.
Without input $K_{b}=0$ and optomagnonic coupling $g=0$, the steady state of
Eq. (7) is the thermal equilibrium state [33] with no photons since
$k_{B}T\ll\hbar\omega_{b}$. The input field operator $\hat{A}_{b}$ of the
propagating photons in the coupler [32; 19; 20] satisfies the commutation
relations
$\left[\hat{A}_{b}(t),\hat{A}_{b}^{\dagger}(t^{\prime})\right]=\delta(t-t^{\prime})$,
with average

$\left\langle\hat{A}_{b}(t)\right\rangle=\sqrt{\frac{P_{b}}{\hbar\omega_{b}}}e^{i\mathcal{W}_{b}(t)},$
(8)

and correlator

$\left\langle\hat{A}_{b}^{\dagger}(t^{\prime})\hat{A}_{b}(t)\right\rangle=\frac{P_{b}}{\hbar\omega_{b}}e^{i(\mathcal{W}_{b}(t)-\mathcal{W}_{b}(t^{\prime}))}.$
(9)

The photons suffer from phase noise that we model by a classical random walk
$\mathcal{W}_{b}(t)=\sqrt{\kappa_{\mathrm{ph}}}\int_{0}^{t}\mathcal{N}(x)dx,$
with dephasing rate $\sqrt{\kappa_{\mathrm{ph}}}$.
$\kappa_{\mathrm{ph}}/(2\pi)$ typically ranges from Hz to MHz [34], much
smaller than the typical inverse lifetimes in a resonator
$\kappa_{\mathrm{ph}}\ll\kappa_{b}\sim 2\pi\times 0.1-1\,$GHz. The phase noise
is taken to be white with
$\left\langle\mathcal{N}\right\rangle_{\mathrm{cl}}=0$ and
$\left\langle\mathcal{N}(t)\mathcal{N}(t^{\prime})\right\rangle_{\mathrm{cl}}=\delta(t-t^{\prime})$.

Since Eq. (7) is linear,
$\hat{W}_{b}(t)=\hat{W}_{b,\mathrm{opt}}(t)+\hat{W}_{b,\mathrm{om}}(t)$, with
optical contribution at large times $t\gg 1/\kappa_{b}$ being,

$\displaystyle\hat{W}_{b,\mathrm{opt}}(t)$
$\displaystyle=-\int_{0}^{t}e^{-(\kappa_{b}+K_{b})(t-\tau)/2}$
$\displaystyle\mathbin{\phantom{=}}\left[\sqrt{\kappa_{b}}\
\hat{b}_{b}(\tau)+\sqrt{K_{b}}\ \hat{A}_{b}(\tau)\right]d\tau$ (10)

includes the thermal noise and input from the coupler. In the steady state and
for $\kappa_{\mathrm{ph}}\ll\kappa_{b}$, we get the commutation relations

$\left[\hat{W}_{b,\mathrm{opt}}\left(t\right),\hat{W}_{b,\mathrm{opt}}^{\dagger}\left(t^{\prime}\right)\right]=e^{-(\kappa_{b}+K_{b})\left|t-t^{\prime}\right|/2},$
(11)

the average

$\left\langle\hat{W}_{b,\mathrm{opt}}(t)\right\rangle=\sqrt{N_{b}}e^{i\mathcal{W}_{b}(t)},$
(12)

and correlator

$\left\langle\hat{W}_{b,\mathrm{opt}}^{\dagger}(t^{\prime})\hat{W}_{b,\mathrm{opt}}(t)\right\rangle=N_{b}e^{i(\mathcal{W}_{b}(t)-\mathcal{W}_{b}(t^{\prime}))}$
(13)

with $N_{b}$ from Eq. (1). The optomagnonic scattering $W_{r}+M\rightarrow
W_{b}$ contributes

$\hat{W}_{b,\mathrm{om}}(t)=-ig\int_{0}^{t}e^{-(\kappa_{b}+K_{b})(t-\tau)/2}\hat{W}_{r}(\tau)\hat{M}(\tau)e^{i\Delta\tau}d\tau.$
(14)

For the red sideband
$\hat{W}_{r}(t)=\hat{W}_{r,\mathrm{opt}}(t)+\hat{W}_{r,\mathrm{om}}(t),$ with
$\hat{W}_{r,\mathrm{opt}}(t)$ analogous to Eq. (10) and scattering
contribution

$\hat{W}_{r,\mathrm{om}}(t)=-ig^{\ast}\int_{0}^{t}e^{-(\kappa_{r}+K_{r})(t-\tau)/2}\hat{W}_{b}(\tau)\hat{M}^{\dagger}(\tau)e^{-i\Delta\tau}d\tau.$
(15)

The magnon envelope operator
$\hat{M}(t)\overset{\triangle}{=}\hat{m}(t)e^{i\omega_{m}t}$ obeys

$\frac{d\hat{M}}{dt}=-ig^{\ast}\hat{W}_{r}^{\dagger}\hat{W}_{b}e^{-i\Delta
t}-\frac{\kappa_{m}}{2}\hat{M}-\sqrt{\kappa_{m}}\hat{b}_{m},$ (16)

where the stochastic magnetic field, $\hat{b}_{m}(t)$, is generated by magnon-
phonon [35], magnon-magnon [36; 37], surface roughness [38] and (rare earth)
impurity scattering [39; 40; 41; 42]. When $k_{B}T/\hbar\gg\kappa_{m}$, which
for $\kappa_{m}\sim 2\pi\times 1\,$MHz [21; 22; 23] means $T\gg
50\,\mathrm{\mu}$K, we can write $\left\langle\hat{b}_{m}(t)\right\rangle=0$,
$\left\langle\hat{b}_{m}^{\dagger}(t^{\prime})\hat{b}_{m}(t)\right\rangle=n_{\mathrm{eq}}\delta(t-t^{\prime})$
and
$\left\langle\hat{b}_{m}(t^{\prime})\hat{b}_{m}^{\dagger}(t)\right\rangle=(n_{\mathrm{eq}}+1)\delta(t-t^{\prime})$,
with average magnon number $n_{\mathrm{eq}}$ [see Eq. (3)]. When $g=0$, the
steady state of Eq. (16) is the Planck distribution of the magnon number at
temperature $T$ [33], given in Eq. (3).

The optical torque $\propto g^{\ast}$ in Eq. (16) generates coherent magnons.
To leading order in $g/\kappa_{r,b}$,

$\frac{d\left\langle\hat{M}\right\rangle}{dt}=-i\bar{\omega}\left\langle\hat{M}\right\rangle-
ig^{\ast}\sqrt{N_{r}N_{b}}e^{-i\Delta
t+i\mathcal{W}(t)}-\frac{\kappa_{\mathrm{eff}}}{2}\left\langle\hat{M}\right\rangle,$
(17)

where

$\bar{\omega}=\left|g\right|^{2}\Delta\left(\frac{4N_{b}}{4\Delta^{2}+\left(\kappa_{r}+K_{r}\right)^{2}}-\frac{4N_{r}}{4\Delta^{2}+\left(\kappa_{b}+K_{b}\right)^{2}}\right),$
(18)

is a shift in the magnon frequency,
$\mathcal{W}=\mathcal{W}_{b}-\mathcal{W}_{r}$ is the phase noise with variance
$2\kappa_{\mathrm{ph}}$, and the effective damping
$\kappa_{\mathrm{eff}}=\kappa_{m}+\bar{\kappa}_{b}-\bar{\kappa}_{r}.$ Here

$\bar{\kappa}_{b}=\frac{4\left|g\right|^{2}N_{r}\left(\kappa_{b}+K_{b}\right)}{4\Delta^{2}+\left(\kappa_{b}+K_{b}\right)^{2}}$
(19)

is proportional to the reaction rate of $W_{r}+M\rightarrow W_{b}$ [see Eq.
(2)] and $\bar{\kappa}_{r}$ is given by $r\leftrightarrow b$. Eq. (17) leads
to the steady state

$\lim_{t\rightarrow\infty}\left\langle\hat{M}(t)\right\rangle=\frac{-ig^{\ast}\sqrt{N_{r}N_{b}}}{i(\bar{\omega}-\Delta)+\kappa_{\mathrm{ph}}+\kappa_{\mathrm{eff}}/2}e^{-i\Delta
t+i\mathcal{W}(t)},$ (20)

where we assumed ergodicity of $\mathcal{W}$. The phase noise of the input
laser fields is imprinted on the magnon amplitude.

We estimate the magnitude of the effects for an input laser with typical
vacuum wavelength $\sim 1\,\mathrm{\mu}$m and
$\omega_{r}\approx\omega_{b}\approx\omega_{\mathrm{opt}}=2\pi\times 300\,$THz.
For a YIG sphere, the optical quality can be as high as
$\omega_{r}/\kappa_{r}=\omega_{b}/\kappa_{b}=10^{6}$ [12] and is limited by
light absorption (for frequencies at which the magneto-optical coupling is
significant). The magnon linewidth $\kappa_{m}=2\pi\times 1\,$MHz and we adopt
the optomagnonic coupling $\left|g\right|=2\pi\times 200\,$Hz [18] for a
sphere of radius $R=300\,\mathrm{\mu}$m (with $\left|g\right|\propto 1/R$). We
assume low phase noise $\kappa_{\mathrm{ph}}\ll\kappa_{m}$ which can otherwise
be absorbed into $\kappa_{\mathrm{eff}}$, cf. Eq. (20). An external magnetic
field can tune $\omega_{m}$ into resonance at $\Delta=0$. For impedance-
matched optical coupling $\kappa_{r,b}=K_{r,b}=\kappa_{\mathrm{opt}}$, the
total magnetic damping

$\kappa_{\mathrm{eff}}=\kappa_{m}\left(1+\frac{P_{r}-P_{b}}{P_{\mathrm{sat}}}\right)$
(21)

with saturation power (to be interpreted below)

$P_{\mathrm{sat}}=\frac{\hbar\kappa_{m}\omega_{\mathrm{opt}}\kappa_{\mathrm{opt}}^{2}}{2\left|g\right|^{2}}=1\,\mathrm{W}.$
(22)

For moderate $P_{r,b}\sim 1-10\,$mW, $\kappa_{\mathrm{eff}}\approx\kappa_{m}$
is limited by the intrinsic (Gilbert) damping of the magnet. For the large
coupling $\left|g\right|=2\pi\times 4\,$kHz predicted for a magnetic vortex in
a thin magnetic disk [17], $P_{\mathrm{sat}}=3.5\,$mW.

Our main result is the number of coherently excited magnons

$n_{\mathrm{c}}=\lim_{t\rightarrow\infty}\left|\left\langle\hat{M}(t)\right\rangle\right|^{2}=\frac{P_{r}P_{b}}{P_{\mathrm{crit}}^{2}},$
(23)

in terms of the critical power

$P_{\mathrm{crit}}=\frac{\hbar\kappa_{\mathrm{eff}}\omega_{\mathrm{opt}}\kappa_{\mathrm{opt}}}{2\left|g\right|},$
(24)

which is a measure for the input power required to generate significant
coherent dyamics. It is smaller than $P_{\mathrm{sat}}$ by a factor
$\kappa_{\mathrm{opt}}/\left|g\right|\sim 10^{6}$. With
$\kappa_{\mathrm{eff}}\approx\kappa_{m}$, $P_{\mathrm{crit}}=1\,\mathrm{\mu}$W
is in experimental reach. We predict a large $n_{c}=10^{6}-10^{8}$ for
$P_{r,b}\sim 1-10\,\mathrm{mW}$. In a magnetic vortex [17],
$P_{\mathrm{crit}}=50\,$nW and $n_{c}=5\times(10^{8}-10^{10})$.

Next we demonstrate that the coherence of the excited magnons is very high (in
the absence of absorption heating by the lasers), i.e. the fluctuations around
the coherent component $\delta\hat{M}=\hat{M}-\langle\hat{M}\rangle$ are very
small, by solving Eq. (16). We employ a weak coupling approximation [19] by
expanding up to the leading terms in $\hat{W}_{x,\mathrm{om}}$. When
$\delta\hat{M}$ varies much slower than $\kappa_{r,b}$ (shown a posteriori to
be equivalent to high optical damping $\kappa_{r,b}\gg\kappa_{\mathrm{eff}}$)
we can replace $\delta\hat{M}(\tau)\rightarrow\delta\hat{M}(t)$ in the
expression of photons Eqs. (14,15). Furthermore, we ignore correlations
between photons and magnons beyond second order in $g$, which is equivalent to
replacing photon operators by their mean-field average (see [19] for
intermediate steps). Then Eq. (16) reduces to

$\frac{d}{dt}\delta\hat{M}=-\left(i\bar{\omega}+\frac{\kappa_{\mathrm{eff}}}{2}\right)\delta\hat{M}-\sqrt{\kappa_{\mathrm{eff}}}\hat{b}_{\mathrm{eff}}$
(25)

where $\bar{\omega}$ and $\kappa_{\mathrm{eff}}$ are defined below Eq. (17)
and the cumulative noise

$\sqrt{\kappa_{\mathrm{eff}}}\hat{b}_{\mathrm{eff}}(t)=\sqrt{\kappa_{m}}\hat{b}_{m}(t)\\\
+ig^{\ast}e^{-i\Delta
t}\left(\hat{W}_{r,\mathrm{opt}}^{\dagger}(t)\hat{W}_{b,\mathrm{opt}}(t)-\sqrt{N_{b}N_{r}}e^{i\mathcal{W}(t)}\right).$
(26)

The statistics for $\kappa_{r,b}\gg\kappa_{\mathrm{eff}}$:
$\left\langle\hat{b}_{\mathrm{eff}}\right\rangle=0$,
$\left\langle\hat{b}_{\mathrm{eff}}^{\dagger}(t^{\prime})\hat{b}_{\mathrm{eff}}(t)\right\rangle\approx
n_{\mathrm{th}}\delta(t-t^{\prime})$, and
$\left\langle\hat{b}_{\mathrm{eff}}(t)\hat{b}_{\mathrm{eff}}^{\dagger}(t^{\prime})\right\rangle\approx\left(n_{\mathrm{th}}+1\right)\delta(t-t^{\prime})$,

$n_{\mathrm{th}}=\frac{\kappa_{m}n_{\mathrm{eq}}+\bar{\kappa}_{r}}{\kappa_{\mathrm{eff}}}\rightarrow\frac{n_{\mathrm{eq}}+P_{b}/P_{\mathrm{sat}}}{1+(P_{r}-P_{b})/P_{\mathrm{sat}}},$
(27)

and $\rightarrow$ holds for impedance-matched optical coupling
$\kappa_{r,b}=K_{r,b}$. Eq. (25) is equivalent to the equation of motion for
magnons in equilibrium [Eq. (16) with $g=0$] after substituting
$\omega_{m}\rightarrow\omega_{m}+\bar{\omega}$,
$\kappa_{m}\rightarrow\kappa_{\mathrm{eff}}$, and
$\hat{b}_{m}\rightarrow\hat{b}_{\mathrm{eff}}$. Therefore in the steady state

$\lim_{t\rightarrow\infty}\left\langle\delta\hat{M}^{\dagger}(t)\delta\hat{M}(t)\right\rangle=n_{\mathrm{th}},$
(28)

justifying the notation $n_{\mathrm{th}}$. At $P_{b}-P_{r}=P_{\mathrm{sat}}$,
the magnon damping $\kappa_{\mathrm{eff}}$ vanishes and the magnon number
$n_{\mathrm{th}}$ diverges. The system becomes unstable and magnon non-
linearities should be taken into account [43]. For $T\sim 1\,$K,
$n_{\mathrm{eq}}\sim 10$ and $n_{\mathrm{th}}\sim n_{\mathrm{eq}}$ for
realistic powers $P_{r,b}\ll P_{\mathrm{sat}}$. Thus, $n_{\mathrm{th}}\ll
n_{c}$, i.e. the coherently precessing magnetization is accompanied only by a
small thermal cloud.

A large magnon population increases the BLS scattering cross section [21; 22;
23]: the uniform mode can be observed in BLS by exciting $>10^{12}$ magnons by
microwaves [22] in spite of the small optomagnonic coupling $g<2\pi\times
5\,$Hz. We consider now the enhancement of BLS by the high-momentum mode $M$
that is coherently excited as discussed above. This can be measured by a third
(probe) beam that couples to another optical WGM. Typically, only one of the
sidepeaks dominates [16], with a ratio of scattered to incident (impedance-
matched) photons

$S=\frac{\left|g^{\prime}\right|^{2}\left(n_{c}+n_{\mathrm{th}}\right)}{\kappa_{\mathrm{opt}}^{2}},$
(29)

where $g^{\prime}$ is the coupling of the probe WGMs with the $M$-magnons and
$\kappa_{\mathrm{opt}}$ is a typical optical linewidth. For
$g^{\prime}=2\pi\times 200\,$Hz we require $P_{r,b}=5\,$mW for a signal that
exceeds the noise background $S_{\mathrm{noise}}\sim 10^{-5}$ [22] . A
threefold larger $\\{g,g^{\prime}\\}$ when reducing the radius to
$100\,\mathrm{\mu}$m increases $S$ by two orders of magnitude (because
$n_{c}\propto|g|^{2}$). For thin magnetic disks with
$\left|g\right|=2\pi\times 4\,$kHz [17] $S\sim 1$.

Coherent magnons can also be excited by femtosecond laser pulses with a
frequency spectrum that overlaps with the two WGMs, a process known as
“impulsive stimulated Raman scattering” [44; 14; 26]. Time periodic and phase-
coherent laser pulses (frequency combs) [45; 46] have a spectrum of sharp and
periodic peaks whose period can be tuned to a magnon frequency. These
techniques can achieve high laser intensities, but are less selective.

In summary, we show that two counter-propagating slightly detuned lasers can
excite a large $\sim 10^{6}-10^{8}$ number of coherent magnons with
sub-$\mathrm{\mu}$m wavelengths in a conventional experimental setup of a
proximity-coupled YIG sphere of radius $\sim 300\,\mathrm{\mu}$m. The
consequent enhancement of the BLS cross section makes it experimentally
feasible to observe. The coherent optical excitation of short-wavelength
magnons with high group velocities can serve as an improved interface between
light and spintronic devices in quantum domain.

## References

  * Chumak  _et al._ [2015] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat Phys 11, 453 (2015).
  * Cherepanov  _et al._ [1993] V. Cherepanov, I. Kolokolov, and V. L’vov, Physics Reports 229, 81 (1993).
  * Kajiwara  _et al._ [2010] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010).
  * Cornelissen  _et al._ [2015] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nature Physics 11, 1022 (2015).
  * Soykal and Flatté [2010] Ö. O. Soykal and M. E. Flatté, Phys. Rev. Lett. 104, 077202 (2010).
  * Zhang  _et al._ [2014] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014).
  * Tabuchi  _et al._ [2014] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014).
  * Bai  _et al._ [2015] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Phys. Rev. Lett. 114, 227201 (2015).
  * Kittel [1958] C. Kittel, Phys. Rev. 110, 836 (1958).
  * Kamra  _et al._ [2015] A. Kamra, H. Keshtgar, P. Yan, and G. E. W. Bauer, Phys. Rev. B 91, 104409 (2015).
  * Kikkawa  _et al._ [2016] T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K.-i. Uchida, Z. Qiu, G. E. W. Bauer, and E. Saitoh, Phys. Rev. Lett. 117, 207203 (2016).
  * Zhang  _et al._ [2016a] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Science Advances 2 (2016a).
  * Borovik-Romanov and Kreines [1982] A. Borovik-Romanov and N. Kreines, Physics Reports 81, 351 (1982).
  * Hansteen  _et al._ [2006] F. Hansteen, A. Kimel, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 014421 (2006).
  * Tabuchi  _et al._ [2016] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Comptes Rendus Physique 17, 729 (2016), quantum microwaves / Micro-ondes quantiques.
  * Sharma  _et al._ [2017] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. B 96, 094412 (2017).
  * Graf  _et al._ [2018] J. Graf, H. Pfeifer, F. Marquardt, and S. Viola Kusminskiy, Phys. Rev. B 98, 241406 (2018).
  * Sharma  _et al._ [2019] S. Sharma, B. Z. Rameshti, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. B 99, 214423 (2019).
  * Sharma  _et al._ [2018] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. Lett. 121, 087205 (2018).
  * Bittencourt  _et al._ [2019] V. A. S. V. Bittencourt, V. Feulner, and S. V. Kusminskiy, Phys. Rev. A 100, 013810 (2019).
  * Osada  _et al._ [2016] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Phys. Rev. Lett. 116, 223601 (2016).
  * Zhang  _et al._ [2016b] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016b).
  * Haigh  _et al._ [2016] J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, Phys. Rev. Lett. 117, 133602 (2016).
  * Osada  _et al._ [2018] A. Osada, A. Gloppe, R. Hisatomi, A. Noguchi, R. Yamazaki, M. Nomura, Y. Nakamura, and K. Usami, Phys. Rev. Lett. 120, 133602 (2018).
  * Haigh  _et al._ [2018] J. A. Haigh, N. J. Lambert, S. Sharma, Y. M. Blanter, G. E. W. Bauer, and A. J. Ramsay, Phys. Rev. B 97, 214423 (2018).
  * Kirilyuk  _et al._ [2010] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010).
  * Prince  _et al._ [2017] R. C. Prince, R. R. Frontiera, and E. O. Potma, Chemical Reviews 117, 5070 (2017), pMID: 27966347.
  * Haigh  _et al._ [2015] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baumberg, A. J. Ramsay, A. Nunnenkamp, and A. J. Ferguson, Phys. Rev. A 92, 063845 (2015).
  * Liu  _et al._ [2016] T. Liu, X. Zhang, H. X. Tang, and M. E. Flatté, Phys. Rev. B 94, 060405 (2016).
  * Pantazopoulos  _et al._ [2017] P. A. Pantazopoulos, N. Stefanou, E. Almpanis, and N. Papanikolaou, Phys. Rev. B 96, 104425 (2017).
  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
  * Marquardt  _et al._ [2007] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007).
  * Gardiner and Collett [1985] C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
  * Tur  _et al._ [1985] M. Tur, B. Moslehi, and J. Goodman, Journal of Lightwave Technology 3, 20 (1985).
  * Streib  _et al._ [2019] S. Streib, N. Vidal-Silva, K. Shen, and G. E. W. Bauer, Phys. Rev. B 99, 184442 (2019).
  * Schlömann [1958] E. Schlömann, Journal of Physics and Chemistry of Solids 6, 242 (1958).
  * Kasuya and LeCraw [1961] T. Kasuya and R. C. LeCraw, Phys. Rev. Lett. 6, 223 (1961).
  * Yu  _et al._ [2019] T. Yu, S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. B 99, 174402 (2019).
  * Sparks  _et al._ [1961] M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791 (1961).
  * Boventer  _et al._ [2018] I. Boventer, M. Pfirrmann, J. Krause, Y. Schön, M. Kläui, and M. Weides, Phys. Rev. B 97, 184420 (2018).
  * Maier-Flaig  _et al._ [2017] H. Maier-Flaig, S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. Huebl, and S. T. B. Goennenwein, Phys. Rev. B 95, 214423 (2017).
  * Kosen  _et al._ [2019] S. Kosen, A. F. van Loo, D. A. Bozhko, L. Mihalceanu, and A. D. Karenowska, arXiv (2019), 1903.02527 [cond-mat.mes-hall] .
  * Viola Kusminskiy  _et al._ [2016] S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, Phys. Rev. A 94, 033821 (2016).
  * Hovhannisyan [2003] D. Hovhannisyan, Microwave and Optical Technology Letters 36, 280 (2003).
  * Hänsch [2006] T. W. Hänsch, Rev. Mod. Phys. 78, 1297 (2006).
  * Kippenberg  _et al._ [2018] T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, Science 361 (2018).