# On Hard Episodes in meta-Learning

Samyadeep Basu, Amr Sharaf
Microsoft AI
<EMAIL_ADDRESS>
&Nicolo Fusi
Microsoft Research
<EMAIL_ADDRESS>
&Soheil Feizi
Department of Computer Science
University of Maryland, College Park
<EMAIL_ADDRESS>

###### Abstract

Existing meta-learners primarily focus on improving the average task accuracy
across multiple episodes. Different episodes, however, may vary in hardness
and quality leading to a wide gap in the meta-learner’s performance across
episodes. Understanding this issue is particularly critical in industrial few-
shot settings, where there is limited control over test episodes as they are
typically uploaded by end-users. In this paper, we empirically analyse the
behaviour of meta-learners on episodes of varying hardness across three
standard benchmark datasets: CIFAR-FS, mini-ImageNet, and tiered-ImageNet.
Surprisingly, we observe a wide gap in accuracy of around $50\%$ between the
hardest and easiest episodes across all the standard benchmarks and meta-
learners. We additionally investigate various properties of hard episodes and
highlight their connection to catastrophic forgetting during meta-training. To
address the issue of sub-par performance on hard episodes, we investigate and
benchmark different meta-training strategies based on adversarial training and
curriculum learning. We find that adversarial training strategies are much
more powerful than curriculum learning in improving the prediction performance
on hard episodes.

##  1 Introduction

Humans have a remarkable ability to learn new concepts from very few examples
and generalize effectively to unseen tasks. However, standard deep learning
approaches still lag behind human capabilities in learning from few examples.
For large over-parameterized deep models, learning with general supervision
from only a few examples leads to over-fitting and thus poor generalization.
To circumvent this, the paradigm of few-shot learning (Wang et al., 2020; Fei-
fei et al., 2006; Vinyals et al., 2017) aims to effectively learn new concepts
from very few labeled examples. These learned concepts can generalize well to
future unseen learning tasks. Several frameworks have been proposed for
tackling the few-shot learning scenario: transfer-learning (Dhillon et al.,
2019), self-training (Phoo & Hariharan, 2020) and meta-learning (Hospedales et
al., 2020; Finn et al., 2017; Snell et al., 2017). Meta-learning in particular
aims to learn the process of learning from few examples and has shown
remarkable performance across various few-shot benchmarks (Hospedales et al.,
2020). In meta-learning, several few-shot tasks (episodes) are sampled from a
set of base classes and the underlying model is trained to perform well on
these tasks leading to improved generalization in learning from only few
examples belonging to novel and unseen classes.

Existing meta-learners such as prototypical networks (Snell et al., 2017),
MAML (Finn et al., 2017), MetaOptNet (Lee et al., 2019), and R2D2 (Bertinetto
et al., 2018) primarily focus on improving prediction performance on average
across multiple episodes. However, different episodes have distinct
characteristics and hardness which might lead to a wide variance in prediction
accuracy across episodes. This problem is much more prevalent in few-shot
models deployed in the industry. For example, meta-trained models are often
deployed in the cloud for the end-users to use for various tasks such as
object recognition, detection, semantic segmentation in computer vision and
natural language understanding in NLP. In such settings, the end-users upload
their own few-shot dataset to perform predictions on new and unseen examples
belonging to novel classes. In practice, different users may upload few-shot
datasets of varying quality and hardness, leading to a wide disparity in
performance across different users. To draw a parallel to the widely accepted
experimental protocols in meta-learning, each of the uploaded few-shot dataset
and the corresponding unseen examples is equivalent to a test episode.

In this paper, we study this issue and investigate how existing state-of-the-
art meta-learners (Snell et al., 2017; Bertinetto et al., 2018; Lee et al.,
2019) perform on episodes of varying hardness. Across three benchmark
datasets: CIFAR-FS, mini-ImageNet, and tieredImageNet, we observe that there
is a gap of $\approx 50\%$ in prediction accuracy between the easiest and
hardest episodes. To this end, we identify several intriguing properties of
hard episodes in meta-learning. For instance, we find that hard episodes are
forgotten more easily than easy episodes during meta-training. To improve
prediction performance on hard episodes, we investigate and benchmark various
adversarial training and curriculum learning strategies that can be used
jointly with any existing meta-learner. Empirically, we find that adversarial
training strategies are much more powerful than curriculum learning in
improving the prediction performance on hard episodes. The aim of our paper is
not to chase another state-of-the-art in meta-learning, but to perform a fine-
grained inspection of hard episodes across various meta-learning methods.

In summary, we make the following contributions:

  * •

We present a detailed analysis of episode hardness in meta-learning across
few-shot benchmarks and state-of-the-art meta-learners. In particular, we
study various properties (e.g., semantic characteristics, forgetting) of
episode hardness across different meta-learners and architectures.

  * •

We find strong connections between episode hardness and catastrophic
forgetting in meta-learning. While catastrophic forgetting can occur when
meta-training with multiple datasets in sequence (Yap et al., 2020), we
observe that forgetting events can occur even when the tasks during meta-
training are drawn from a single dataset. In particular, we find that hard
episodes are easy to forget, while easy episodes are difficult to forget.

  * •

Based on our analysis, we investigate and benchmark different adversarial
training and curriculum training strategies to augment general purpose meta-
training for improving prediction performance on hard episodes. Empirically,
we find that although there is no one-size-fits-all solution, adversarial
meta-training strategies are more powerful when compared to curriculum
learning strategies.

##  2 Background and Related Work

Meta-learning aims to learn an underlying model that can generalize and adapt
well to examples from unseen classes by the process of learning to learn. This
is primarily achieved by mimicking the evaluation and adaptation procedure
during meta-training. In general, there are three types of meta-learners: (a)
Memory-based methods (Ravi & Larochelle, 2017; Munkhdalai et al., 2018;
Santoro et al., 2016) adapt to novel classes with a memory attached to the
meta-learner; (b) Metric-learning based methods (Snell et al., 2017; Sung et
al., 2017) aim to learn transferable deep representations which can adapt to
unseen classes without any additional fine-tuning; (c) Optimization based
methods (Finn et al., 2017; Lee et al., 2019; Bertinetto et al., 2018) learn a
good pre-training initialization for effective transfer to unseen tasks with
only a few optimization steps. Although the primary focus of our work is meta-
learning, we note that other few-shot learning paradigms such as transfer
learning (Chen et al., 2021; Sun et al., 2019; Dhillon et al., 2020) have also
shown competitive performance with meta-learning.

While there has been a significant progress in improving the state-of-the-art
in meta-learning, very few work investigates the effectiveness of existing
meta-learning approaches on episodes of varying hardness. A recent and
concurrent work by Arnold et al. (2021) discusses episode difficulty and the
impact of random episodic sampling during meta-training. Based on their
analysis, Arnold et al. (2021) propose a re-weighted optimization framework
for meta-training based on importance sampling. Although our paper and Arnold
et al. (2021) tackle similar problems of episodic hardness, there are several
points which distinguishes our work:

  * •

We provide a much more fine-grained analysis of episode hardness than Arnold
et al. (2021). Arnold et al. (2021) primarily discuss the transferability of
episodes across different meta-learners, while we find and investigate a
strong connection between episode hardness and catastrophic forgetting.

  * •

Arnold et al. (2021) propose a loss re-weighting framework for improving the
average accuracy across episodes. In contrary, we investigate the
effectiveness of adversarial training (Gong et al., 2020) and general
curriculum learning techniques in improving the average as well as worst-case
prediction performance in meta-learning.

Adversarial meta-learning techniques have previously been used in conjunction
with data-augmentation (Ni et al., 2021) to select the augmentation type
resulting in the worst-case loss among different augmentation techniques. In
this paper, we focus on how such strategies can be useful in improving the
prediction performance of the hard episodes in addition to the average
accuracy.

Figure 1:  Accuracy (y-axis) of existing meta-learners on the hardest and the
easiest episode across standard few-shot datasets and meta-learners (x-axis).
Note that there is a wide gap of $\approx 50\%$ between the prediction
performance on the easiest and hardest episode.

##  3 Rethinking Episodic Accuracy

Existing state-of-the-art meta-learners (Finn et al., 2017; Lee et al., 2019;
Snell et al., 2017; Bertinetto et al., 2018) primarily focus on optimizing for
the average loss across multiple training episodes or tasks. However, solely
the average performance in isolation does not give enough insights into how
meta-learners perform on episodes of varying quality and hardness. Such
insights can be particularly crucial to investigate and debug meta-learning
models deployed in the wild, where the model can encounter diverse test
episodes. In this section, we go beyond the average accuracy across different
test episodes and evaluate meta-learners on episodes of varying hardness.
First, we discuss how to quantify the hardness of an episode and then discuss
the performance of meta-learners on hard episodes.

###  3.1 What is a good measure of episode hardness?

Episodic sampling (i.e. sampling various few-shot tasks from a base dataset)
in meta-learning takes place in two steps: (i) First the episode classes are
sampled from the class distribution of the base classes : $c\sim
p(\mathcal{C}_{base})$; (ii) Next, an episode $\tau$ is sampled from the data
distribution conditioned on the set of sampled classes $c$: $\tau\sim
p(\mathcal{D}|c)$, where $\mathcal{D}$ is the base dataset. An episode $\tau$
consists of a set of support examples $\tau_{s}$ and query examples
$\tau_{q}$. In few-shot learning, a $n$-way, $k$-shot episode is sampled which
results in sampling $n$ classes and $k$ support examples per class. Based on
this, the meta-learning optimization objective can be generalized as the
following:

$\theta^{*}=\arg\min_{\theta}\mathbb{E}_{\tau}[\ell(\mathcal{F}_{\theta^{{}^{\prime}}},\tau_{q})]$
(1)

where $\mathcal{F}$ is the base architecture with $\theta$ as the model
parameters and $\theta^{{}^{\prime}}=\mathcal{A}(\theta,\tau_{s})$ is the
fine-tuning step with the support examples. Different meta-learners have
different types of fine-tuning procedures and we direct the readers to (Finn
et al., 2017; Snell et al., 2017; Bertinetto et al., 2018) for more
information on the characteristics of $\mathcal{A}$. Based on this definition,
we define the hardness of an episode $\mathcal{H}(\tau)$ in terms of the loss
incurred on the query examples in an episode:

$\mathcal{H}(\tau)=\ell(\mathcal{F}_{\theta^{*}},\tau_{q})$ (2)

We choose query loss as a metric for hardness because of its inherent
simplicity in computation as well as interpretation. In addition, we find a
strong negative correlation between the episodic loss and the accuracy
($\approx-0.92$ for mini-ImageNet and $\approx-0.89$ for tieredImageNet with
prototypical networks). This is true for other meta-learners such as R2D2 too
(See Appendix A for more details). Alternatively, hardness of an episode can
also be defined as the average log-odds of the query example (Dhillon et al.,
2020).

###  3.2 Performance of meta-learners on hard episodes

To understand the effectiveness of meta-learners on episodes of varying
hardness, we first order the test episodes in decreasing order of their
hardness. Then, we evaluate different meta-learners on the easiest and the
hardest test episode. Across all the few-shot benchmark datasets such as mini-
ImageNet, CIFAR-FS and tieredImageNet, we find in Fig. (1) that there is a gap
in accuracy of $\approx 50\%$ between the episodes with the highest and the
lowest loss. Furthermore, we find that the meta-learner and architecture which
performs well on average, does not necessarily perform well on the hardest
episode. For example, in the case of mini-ImageNet, prototypical networks with
a stronger architecture such as ResNet performs better than the 4-layered
convolutional architecture on an average, but not on the hard episodes.
Moreover in Fig. (1), we notice that for tieredImageNet, prototypical networks
with ResNet performs the best on easy episodes, while R2D2 with ResNet
performs slightly better on the hard episodes. The wide gap in accuracy
between hard and easy episodes can be magnified for meta-learners deployed in
the wild, where the model might encounter episodes which are significantly
hard and diverse in nature. Going forward, we believe that meta-learning
studies should not only report average episodic accuracy, but also the
prediction accuracy on easy and hard episodes, to present a complete picture
of the effectiveness of the meta-learner.

##  4 Visual Semantics of Hard Episodes

Based on the observed disparity in performance between hard and easy episodes,
a natural question arises: what causes certain episodes to incur high loss?
While quantitatively, the hardness of an episode can be defined in terms of
the loss incurred on the query examples, it does not provide salient insights
into the qualitative nature of hard episodes. In general, we find that
episodes incur high loss when there is a mismatch in the semantic
characteristics between the support and query examples. For example, when a
majority of the support examples have objects of only one category in the
frame and the query examples have multiple objects of different categories
surrounding the primary object of interest, the underlying meta-learner often
leads to a wrong prediction. Furthermore, when the shape of the objects in the
query examples is slightly different from the objects in the support examples,
the prediction is often erroneous.

Figure 2:  Semantic Properties of Hard and Easy Episodes: (a) Hard episode,
Class: electric-guitar; (b) Easy episode, Class: mixing-bowl; The images
marked in red borders are misclassified query examples.

In Fig. (2)-(a), we notice that the query examples that have different objects
(e.g., humans) along with the primary object (i.e., guitar) are often
misclassified. In Fig. (2)-(b), where most of the query examples are
classified correctly, we find that the misclassified examples are of two
types: (i) query images in which the primary object is occluded with a
secondary object; (ii) the shape of the object in the query example is
different from the object shapes in the support examples. We provide
additional examples of hard episodes in Appendix E.

##  5 Hard Episodes Suffer from Forgetting

In supervised learning, catastrophic forgetting is prevalent when tasks from
different distributions are learned sequentially (Kirkpatrick et al., 2016).
In such cases, old tasks are forgotten as the model encounters and learns from
new tasks. Toneva et al. (2019) has shown that certain examples can be
forgotten with high frequency during the course of supervised training even
when the samples are drawn from a single dataset. In meta-learning, Yap et al.
(2020) has shown that in meta-training with tasks from multiple task
distributions sequentially, tasks from the old distribution in the sequence
can be forgotten as the meta-learner encounters new tasks. However, we observe
that even in the case of meta-training with tasks drawn from a single task
distribution, certain types of tasks (episodes) can be forgotten during the
course of training. In particular, we analyze the connection between the
hardness of episodes and catastrophic forgetting in meta-learning. We track
the behaviour of easy and hard episodes during meta-training and in summary
find that:

(i) For hard episodes, we notice that the final accuracy at the end of the
training drops significantly ($\approx 15\%$ in some cases) from the maximum
accuracy obtained during the course of meta-training.

(ii) Hard episodes have more number of forgetting events in comparison to the
easy episodes during the course of meta-training. This behaviour is more
pronounced in the later stages of meta-training where the accuracies of the
easy episodes have already stabilized.

Figure 3: Accuracy of easy and hard episodes (y-axis) during the course of
meta-training across different epochs (x-axis); Hard episodes often have a
final accuracy less than the maximum accuracy reached during meta-training.

###  5.1 Defining Forgetting Events in meta learning

During the course of meta-training, the set of sampled tasks are different in
each epoch. In order to track forgetting events during meta-training, we first
randomly select a set of $k$ episodes ($\mathcal{E}=\\{\tau\\}_{i=1}^{k}$) and
track their accuracy, throughout the course of meta-training. In our
experiments, we set $k=160$ . We primarily define two types of forgetting
events: (i) Local forgetting event; (ii) Global forgetting event.

Global forgetting events. For a given episode, a global forgetting event is
encountered if the accuracy of the episode at the end of meta-training is less
than the maximum accuracy reached during the course of training by a
particular threshold. Formally, given an episode $\tau$ with the maximum
accuracy $acc_{max}(\tau)=\max_{j}acc_{j}(\tau)$, a global forgetting event
occurs if $acc_{max}(\tau)\geq acc_{end}(\tau)+\alpha$, where $\alpha$ is a
threshold and $acc_{end}(\tau)$ is the accuracy at the end of meta-training.
Note that for each episode, a global forgetting event can occur only once.

Local forgetting events. For an episode $\tau$ in the $j^{th}$ epoch of meta-
training, a local forgetting event is encountered if the accuracy of the
episode at the $j^{th}$ epoch ($acc_{j}(\tau)$) is less than the accuracy at
the $(j-1)^{th}$ epoch ($acc_{j-1}(\tau)$) by a particular threshold, denoted
by $\alpha$. Formally, a local forgotten event is encountered if
$acc_{j}(\tau)+\alpha\leq acc_{j-1}(\tau)$. Empirically, we study local
forgetting events for $0.03\leq\alpha\leq 0.15$.

###  5.2 Forgetting Events and Hard Episodes

Global forgetting events. In Fig. (3), we track the accuracy of the hardest
and easiest episode from each of the few-shot datasets during the entire
course of meta-training across different epochs. Visually, we observe that for
the hard episode, the accuracy decreases after a certain point during the
course of meta-training. However for the easy episode, this is not the case
and the accuracy increases till the end of meta-training. To draw more
insights, we compute the global forgetting behaviour of different episodes. We
first choose the 15 hardest and easiest episodes respectively from CIFAR-FS,
mini-ImageNet, tieredImageNet and compute their final episodic accuracy and
the maximum episodic accuracy reached during meta-training. In Fig. (4), we
find that for the hard episodes, the gap between the final accuracy and the
maximum accuracy reached during meta-training is significantly larger than the
easy episodes. The gap in particular is large for mini-ImageNet and
tieredImageNet, while for CIFAR-FS the gap is relatively narrow. For example,
in case of the mini-ImageNet dataset, the gap can be $\approx 15\%$, whereas
for the tieredImageNet dataset, this gap can be $\approx 10\%$. For CIFAR-FS,
this gap is $\approx 6\%$. This characteristic shows that hard episodes are
globally forgotten in comparison to the easier episodes during the entire
course of meta-training. We provide further results on global forgetting
events in Appendix C.

Figure 4:  Hard episodes have a wider gap between the final accuracy (y-axis)
and the maximum accuracy reached during meta-training (x-axis), in comparison
to easy episodes. This behaviour is more pronounced for mini-ImageNet and
tieredImageNet.

Local forgetting events. In order to compute the frequency of local forgetting
events for easy and hard episodes across the three few-shot datasets, we first
choose 15 easy and hard episodes from each dataset. Across this set of easy
and hard episodes, we then compute the number of local forgetting events
across various thresholds. In general, across the entire duration of meta-
training, we find that the hard episodes have more local forgetting events
than the easy episodes. For instance, in Fig. (5)-(a), we observe that there
is a substantial gap in the number of forgetting events encountered for easy
and hard episodes during the entire course of meta-training. Furthermore, to
gain more insights about this gap in the number of encountered forgetting
events, we understand how this gap behaves in the first 20 epochs of meta-
training (Fig. (5)-(b)) and the last 20 epochs of meta-training (Fig.
(5)-(c)). In particular, we find that the gap is narrow during the initial
stages of meta-training, whereas the gap widens substantially during the later
stages.

To summarize, we find that forgetting occurs in meta-learning even when the
tasks are drawn from a single task distribution. Furthermore, we find a strong
connection between episode hardness and forgetting events, where we show that
hard episodes are more easily forgotten than easy episodes, both in the local
and global contexts. In the next section, we investigate two meta-training
strategies to improve few-shot performance on hard episodes.

Figure 5:  (a) Total number of local forgetting events (y-axis) across
different thresholds (x-axis) during the course of meta-training; (b) Total
number of local forgetting events during the first 20 epochs of meta-training;
(c) Total number of local forgetting events during the last 20 epochs of meta-
training. The number of local forgetting events is higher for hard episodes in
comparison to the easy episodes across different thresholds.

##  6 Improving Performance on Hard Episodes

In this section, we investigate and benchmark two different meta-training
strategies based on adversarial training and curriculum learning in order to
improve prediction performance on hard episodes. Recent work (Gong et al.,
2020; Ni et al., 2021) uses adversarial training to select episode specific
augmentations from a wide pool of diverse data augmentation methods resulting
in the highest loss. This loss is then optimized with respect to the model
parameters during training. Such data augmentation selection strategies have
been shown to mitigate over-fitting and improve generalization. In our work,
we study how such training strategies can be used to select hard episodes and
improve prediction performance across these episodes. In particular, we
investigate two variants of adversarial training as proposed first in Gong et
al. (2020): (i) General adversarial training (AT); (ii) Adversarial curriculum
training (ACT).

###  6.1 General adversarial training

We adopt the adversarial training procedure from (Gong et al., 2020) to first
select episodes with a high loss and optimize the meta-learner only with
respect to the loss incurred by such hard episodes. Specifically, this
involves solving a saddle-point optimization problem where the underlying loss
is minimized with respect to the parameters of the model and maximized with
respect to the input. In particular, during each update of the meta-learner,
we first draw a batch of episodes each containing support and query examples.
Then for each element of the batch, we additionally sample a number of similar
episodes and select the episode with the highest loss. The model parameters
are then updated with respect to the gradient of the loss incurred by the
selected hard episode. Formally, we solve the following min-max optimization:

$\min_{\theta}\mathbb{E}_{\tau}[\max_{t\in
g(\mathcal{\tau})}\ell(\mathcal{F}_{\theta^{{}^{\prime}}},t_{q})]$ (3)

where $\theta^{{}^{\prime}}=\mathcal{A}(\theta,t_{s})$ is the fine-tuning step
with the support examples from the selected episodes with the highest loss,
$g(\tau)$ is an operator which samples additional episodes to select from for
each task $\tau$ in the batch, $\mathcal{F}$ is the base learner and $\ell$ is
the loss function. In our experiments, we let $g(\tau)$ select four additional
episodes per sampled episode $\tau$. We provide a more detailed description of
the hyper-parameters in Appendix B.1.

###  6.2 Adversarial Curriculum Training

Curriculum learning (Hacohen & Weinshall, 2019; Bengio et al., 2009) aims to
mimic the learning process of humans and animals. In particular, humans learn
new tasks in a well defined order; i.e., it first learns easy tasks and then
gradually moves towards learning more complex tasks. Inspired by this, we
modify Eq. (3) and investigate a curriculum meta-training strategy.
Specifically, during the initial phase of meta-training, we select only a set
of easy episodes to learn from, while in the later stages of training, harder
episodes are selected. Formally, given the underlying model is meta-trained
for $|e|$ epochs, for the first $|e|/2$ epochs, the following loss is
optimized:

$\min_{\theta}\mathbb{E}_{\tau}[\min_{t\in
g(\mathcal{\tau})}\ell(\mathcal{F}_{\theta^{{}^{\prime}}},t_{q})]$ (4)

and for the last $|e|/2$ epochs during meta-training, the general min-max
adversarial loss described in Eq. (3) is used.

##  7 Experiments

###  7.1 Experimental Setup

Datasets. We use three standard few-shot classification datasets for our
experiments : (i) CIFAR-FS (Bertinetto et al., 2018); (ii) mini-ImageNet
(Vinyals et al., 2016) and (iii) tieredImageNet (Ren et al., 2018). CIFAR-FS
is sampled from CIFAR-100, whereas mini-ImageNet and tieredImageNet are
subsets of ImageNet (Deng et al., 2009). We use the class splits from (Lee et
al., 2019) for all the three datasets. Note that tieredImageNet is a more
challenging dataset than mini-ImageNet as the splits are constructed from near
the root of the ImageNet hierarchy. We provide more details on the datasets in
Appendix B.2.

Method | CIFAR-FS | mini-ImageNet | tieredImageNet
---|---|---|---
| 1-shot | 5-shot | 1-shot | 5-shot | 1-shot | 5-shot
| Conv-ProtoNet
---
\+ AT
\+ ACT
| 62.6
---
63.3
63.1
| 80.9
---
80.8
80.0
| 52.3
---
52.7
52.4
| 70.4
---
70.8
70.3
| 52.6
---
52.7
52.9
| 71.5
---
71.9
71.5
| ResNet12-ProtoNet
---
\+ AT
\+ ACT
| 71.0
---
71.2
69.3
| 84.0
---
83.9
82.6
| 59.0
---
58.9
56.0
| 74.8
---
74.9
72.8
| 61.8
---
62.9
63.4
| 80.0
---
80.4
79.9
| Conv-R2D2
---
+AT
+ACT
| 67.6
---
68.0
68.3
| 82.6
---
82.8
82.6
| 55.3
---
55.7
55.4
| 72.4
---
72.4
71.3
| 56.8
---
57.2
57.8
| 75.00
---
75.00
74.9
| ResNet12-R2D2
---
+AT
+ACT
| 70.0
---
71.0
69.4
| 84.8
---
84.9
83.2
| 58.6
---
58.63
56.3
| 75.5
---
76.0
74.4
| 62.8
---
63.7
63.5
| 80.4
---
81.3
80.9
| ResNet12 -MetaOptNet
---
+AT
+ACT
| 70.8
---
71.2
70.2
| 84.0
---
84.6
83.9
| 60.1
---
60.9
59.8
| 77.4
---
78.2
77.1
| 62.9
---
63.3
62.1
| 80.7
---
80.9
79.8
Table 1: Average episodic performance of general adversarial training (AT) and
adversarial curriculum training (ACT) across different meta-learners.

Architectures. We primarily use two standard architectures for our
experiments: (i) 4-layer convolutional network as introduced in (Vinyals et
al., 2017); (ii) ResNet-12 (He et al., 2015) which is used by (Oreshkin et
al., 2018) in the few-shot setting. Both these architectures use batch-
normalization after every convolutional layer and use ReLU as the activation
function. Similar architectures for few-shot learning have been previously
used in (Lee et al., 2019; Arnold et al., 2021).

Meta-learners. We use prototypical networks (Snell et al., 2017) from the
metric learning family of few-shot algorithms. In addition, we use MetaOptNet
(Lee et al., 2019) and R2D2 (Bertinetto et al., 2018) as representative
algorithms from the optimization based meta-learning methods.

Model Selection. We use the validation set for each dataset to select the best
model. Primarily, we run the validation procedure every 1k iterations on 2k
episodes from the validation set to select the best model. Finally, we
evaluate on 1k test episodes from the test set for each dataset.

###  7.2 Discussion

Method | CIFAR-FS | mini-ImageNet | tieredImageNet
---|---|---|---
| 1-shot | 5-shot | 1-shot | 5-shot | 1-shot | 5-shot
| Conv-ProtoNet
---
\+ AT
\+ ACT
| 28.8
---
29.3
28.5
| 63.0
---
62.9
61.3
| 20.2
---
19.3
19.4
| 53.2
---
53.5
52.7
| 22.2
---
24.6
22.1
| 52.3
---
52.7
53.1
| ResNet12-ProtoNet
---
\+ AT
\+ ACT
| 36.0
---
33.7
29.46
| 66.7
---
66.7
64.9
| 26.2
---
29.2
24.5
| 56.5
---
58.3
54.7
| 29.8
---
31.2
29.6
| 60.4
---
60.9
60.4
| Conv-R2D2
---
+AT
+ACT
| 32.2
---
35.3
32.8
| 65.3
---
65.7
65.5
| 21.4
---
24.0
23.7
| 55.7
---
55.6
54.5
| 27.0
---
27.6
27.8
| 56.6
---
56.6
56.7
| ResNet12-R2D2
---
+AT
+ACT
| 30.6
---
35.6
31.8
| 68.0
---
68.0
65.9
| 28.0
---
28.5
24.4
| 59.0
---
60.1
56.8
| 34.0
---
33.8
30.8
| 61.5
---
62.1
62.0
| ResNet12 -MetaOptNet
---
+AT
+ACT
| 37.2
---
37.9
37.1
| 69.2
---
70.0
69.1
| 29.5
---
31.2
29.2
| 61.1
---
62.3
60.5
| 30.4
---
31.5
30.2
| 61.5
---
61.4
60.9
Table 2:  Performance of general adversarial training (AT) and adversarial
curriculum training (ACT) across different meta-learners on hard episodes. We
report the mean accuracy over 30 hardest episodes for each meta-learner.

####  7.2.1 Adversarial training improves performance on hard episodes

Across different meta-learners (ProtoNets, R2D2, MetaOptNet) and few-shot
datasets (CIFAR-FS, mini-ImageNet, tieredImageNet), we find that the
adversarial training (AT) strategy works well in general to improve both the
average episodic performance as well as the episodic performance on hard
episodes. The comprehensive results for the average performance of meta-
learners is presented in Table (1), while the results on hard episodes are
presented in Table (2). In particular, we find that adversarial meta-training
strategies never hurt the average episodic performance and improves over the
baseline in a majority of our experimental settings. However, we find that the
adversarial training strategy (AT) leads to a large gain over the baseline
meta-training strategy for hard episodes. Specifically, we find that the
improvements are more significant for the 1-shot case when compared to the
5-shot case. For example. specific to the 1-shot case, we observe a $5\%$ gain
for CIFAR-FS with R2-D2 and $3\%$ gain for mini-ImageNet with prototypical
networks. For tieredImageNet, we observe $\approx 2\%$ improvement on episodic
performance for hard episodes with prototypical networks.

####  7.2.2 Adversarial training is better than curriculum training

Although curriculum training leads to better generalization in supervised
learning (Hacohen & Weinshall, 2019; Bengio et al., 2009), we find that in
meta-learning, the adversarial curriculum strategy (ACT) generally performs
worse than both the baseline and general adversarial training (AT) in a
majority of our experimental settings. Our observation on curriculum training
for meta-learning is consistent with the recent work of (Arnold et al., 2021)
where they show that curriculum meta-training strategies underperform
significantly when compared to the baseline meta-training. We note that
although the curriculum formulation in (Arnold et al., 2021) is different than
ours, both methods present easy episodes to the meta-learner first followed by
hard episodes. While we present a negative result on curriculum meta-training,
we believe that this observation can be used as a note to develop more
advanced and improved curriculum meta-training strategies in the future.

In summary, we find that although there is no one-size-fits-all solution to
improve performance on hard episodes, adversarial meta-training strategies
perform better than the baseline and curriculum learning.

##  8 Conclusion

In our paper, we investigated how different meta-learners perform on episodes
of varying hardness. We found that there exists a wide gap in the performance
of the meta-learners between the easiest and the hardest episode across
different few-shot datasets. Furthermore, we investigated various facets of
the hard episodes and uncovered two major properties: (i) Hard episodes
usually have multiple diverse objects in the query examples, whereas the
support set primarily consists of objects of a single category, (ii) Hard
episodes are forgotten by the underlying meta-learner at a higher frequency
than the easier episodes. To improve prediction performance over hard
episodes, we investigated and benchmarked different training strategies such
as adversarial training and curriculum learning during meta-training. We found
that adversarial training strategies are beneficial towards meta-training and
improve the average episodic performance as well as performance over hard
episodes. Based on our analysis in this paper, designing more robust meta-
learning algorithms which can generalize to hard episodes is an important
direction of future work.

##  9 Reproducibility Statement

Small experimental details are crucial to reproduce results in deep learning.
In order to foster the reproducibility of our paper, we provide all the small
details of our experiments in the Appendix (Section : Hyperparameters). We
would like to point out that we use the general hyperparameters in our
experimental setup, as used in (Lee et al., 2019) to ensure reproducibility of
the baselines which is particularly important in few-shot learning research.
Additionally, considering a majority of our paper is on analysis of meta-
learners and generating plots, we provide all the necessary details (e.g,
number of episodes chosen for generating the plots) available in each
corresponding section (See the subsections of Section 5).

##  10 Ethics Statement

In recent times, there has been a significant increase in the usage of deep
learning models in the industry. Domains of application include NLP, computer
vision and healthcare to name a few. Training task and domain specific deep
models require large amounts of labeled training data which is not available
for a variety of use-cases. The paradigm of few-shot learning (or meta-
learning) aims to learn generalizable models from only a few training
examples. Our paper is primarily geared towards understanding the science of
such few-shot learning methods. We believe our work can accelerate research on
few-shot learning with a specific focus on improving learning from hard
episodes. To the best of our knowledge our work will facilitate few-shot
learning research and does not lead to any negative or unethical societal
impact.

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##  Appendix A Loss as a measure of hardness

Figure 6:  Loss (x-axis) vs. Accuracy (y-axis) plot for miniImageNet. Figure
7:  Loss (x-axis) vs. Accuracy (y-axis) plot for CIFAR-FS. Figure 8:  Loss
(x-axis) vs. Accuracy (y-axis) plot for tieredImageNet. Figure 9:
Distribution of episode hardness (difficulty) for mini-ImageNet across
different meta-learners. We find that the distribution of hardness
approximately follows a gamma distribution. From the density plots, we find
that a significant fraction of the episodes lie in the range of medium-hard
range of difficulty.

##  Appendix B Implementation Details

###  B.1 Hyperparameters

Similar to (Lee et al., 2019), for the optimizer we use SGD with Nesterov
momentum and weight decay. We set the momentum to be 0.9 and weight decay to
be 0.0005. During meta-training, the underlying meta-learner is trained for 60
epochs with each epoch consisting of 1000 episodes. We use 8 episodes per
batch during the meta-learner update. The initial learning rate is kept at 0.1
which is then decayed using a learning rate scheduler at epoch 20, 40 and 50
as shown in (Lee et al., 2019). We use a 5-way classification for each of our
experimental settings. During each iteration of meta-training, we use 6 query
examples, while during meta-testing we use 15 query examples. Following the
practice of (Gidaris & Komodakis, 2018; Qiao et al., 2017; Lee et al., 2019),
we use horizontal flip, random crop, and color jitter as data augmentation
techniques during meta-training. Across each of the experimental setup, the
best model for meta-testing is chosen based on the best validation set
performance. In general, few-shot learning can be performed in two scenarios:
(i) Inductive setting where each test example is evaluated independently; (ii)
Transductive setting where the few-shot learner has access to all the test
examples. In all our experimental settings, we follow the inductive setting
where each query example is evaluated independently. For general adversarial
training method (AT), we choose sample 4 additional episodes per sampled
episode. For a batch of size 8, there would be 32 additional episodes with a
total of 40 episodes. 8 episodes with the highest loss are selected from this
pool for optimizing the meta-learner. For our adversarial curriculum learning
method (ACT), we choose the set of 8 episodes with the lowest loss for the
first 30 epochs of meta-training and choose the set of 8 episodes during the
last 30 epochs of meta-training. Across all our experimental settings, we use
the same number of shots during meta-training and meta-testing to match the
training and testing conditions.

Figure 10:  Accuracy of hard and easy episodes (y-axis) during the course of
meta-training across different epochs (x-axis). Different colors signify
different episodes.

###  B.2 Few-shot Datasets

mini-ImageNet .  The mini-ImageNet dataset is introduced by (Vinyals et al.,
2016) and is a standard few-shot classification benchmark. This dataset
consists of 100 classes which are chosen from ILSVRC-2012 (Deng et al., 2009).
The 100 classes are split into 64, 16, and 20 classes for meta-training, meta-
validation and meta-testing. Each class has 600 images resulting in a total of
60000 images for the entire dataset.

tieredImageNet. The tieredImageNet dataset is a larger and more challenging
few-shot learning benchmark. It consists of 608 classes in total which are
subclasses of ILSVRC-2012 (Deng et al., 2009). The total number of classes are
split as 351, 97 and 160 classes for meta-training, meta-validation and meta-
testing respectively. These classes are selected such that there is minimum
semantic similarity between the different splits, which makes this dataset
challenging.

CIFAR-FS. The CIFAR-FS dataset (Bertinetto et al., 2018) is curated from
CIFAR-100 and comprises of 100 classes in total with each class consisting of
600 images. The classes are randomly split into 64, 16 and 20 for meta-
training, meta-validation and meta-testing respectively.

Amongst the three datasets, the images are of size 84x84 for mini-ImageNet and
tieredImageNet, while for CIFAR-FS, the images are of size 32x32.

##  Appendix C Global Forgetting Events

To gain further insights into global forgetting events, we select 5 hard and
easy episodes from each of the few-shot datasets and track their accuracy
during the course of meta-training. We use prototypical networks as the base
meta-learner for our analysis. We find and validate in Fig. (10), that our
analysis on global forgetting (as shown in the main paper) holds true for more
number of hard episodes. Additionally, we also choose 25 hard and easy
episodes to compute the mean of their maximum accuracy reached during meta-
training and the accuracy at the end of training. We find that for hard
episodes on an average, the gap between the maximum accuracy reached during
training and the final accuracy is large. However, for easy episodes we find
this gap to be narrow. This large gap for hard episodes signifies they they
undergo catastrophic forgetting during the course of meta-training.

Figure 11:  For a set of 25 hard and easy episodes, we report the mean of the
maximum accuracy reached during the course of meta-training and the accuracy
at the end of meta-training. For hard episodes, we observe a substantial gap
denoting global forgetting.

##  Appendix D On Transferability of Episodes

Recent and concurrent work such as (Arnold et al., 2021) shows that episode
hardness transfer across different architectures and meta-learners. In this
section, we revisit their analysis and take a closer look at the
transferability of episodes across various meta-learners and architectures to
obtain certain new insights. To this end, we compute the Pearson and Spearman
correlations of the losses (hardness) incurred by episodes across different
architectures and meta-learners for each few-shot dataset.

Figure 12:  Transferability of episode hardness for tieredImageNet. Figure 13:
Transferability of episode hardness for mini-ImageNet. Figure 14:
Transferability of episode hardness for CIFAR-FS.

Across all the three few-shot datasets, we find that the order of
transferability across meta-learners and architectures is similar. For
example, episodes transfer the best consistently between ResNet-MetaOptNet and
ResNet-R2D2 across all the datasets, while episodes transfer the worst between
ResNet-R2D2 and Conv-Proto. This observation makes intuitive sense as the
meta-training strategies for both R2D2 and MetaOptNet are more similar for
than R2D2 and prototypical networks. Amongst the datasets, we find that
transfer of episode hardness between meta-learners and architectures is much
more effective for tieredImageNet than mini-ImageNet. For example, out of the
7 combinations for transferability of episodes, 3 of them have a correlation
of more than 0.90 in case of tieredImageNet.

##  Appendix E More on Visual Semantics of Hard Episodes

Figure 15: Visual semantic properties of hard episodes: We find that if there
are closely related fine-grained categories in an episode (e.g, Malamute vs.
Dalmatian), the meta-learner (prototypical networks + ResNet-12) often gives a
wrong prediction.

In this section, we provide more instances of hard episodes. In particular, we
find that episodes also become hard when there are closely related fine-
grained categories in the episode. For example, in an episode consisting of
dogs from the classes Malamute and Dalmatian, we notice that the underlying
meta-learner often gets confused between the two fine-grained categories
leading to erroneous predictions on the query examples. Presence of similar
fine-grained categories in episodes is therefore one failure mode of existing
meta-learning methods.


# On Kazhdan-Yom Din asymptotic Schur orthogonality for K-finite matrix
coefficients

Anne-Marie Aubert, Alfio Fabio La Rosa

###### Abstract

In a recent article, D. Kazhdan and A. Yom Din conjectured the validity of an
asymptotic form of Schur’s orthogonality for tempered irreducible unitary
representations of semisimple groups defined over local fields. In the non-
Archimedean case, they established such an orthogonality for $K$-finite matrix
coefficients. Building on their work, and exploiting the admissibility of
irreducible unitary representations, we prove the analogous result in the
Archimedean case.

###### Contents

  1. 1 Introduction
  2. 2 Recollections on $(\mathfrak{g},K)$-modules
  3. 3 Schur orthogonality

##  1 Introduction

Let $G$ be a semisimple group over a local field, let $K$ be a maximal compact
subgroup of $G$. We fix a Haar measure on $G$, denoted $\,dg$. If $H$ is the
Hilbert space underlying a unitary representation of $G$, let $H_{K}$ denote
the space of $K$-finite vectors and $H^{\infty}$ the space of smooth vectors.

In their recent work [4], D. Kazhdan and A. Yom Din conjectured the validity
of an asymptotic version of Schur’s orthogonality relations. It should hold
for matrix coefficients of tempered irreducible unitary representations of
$G$, generalising Schur’s well-known orthogonality relations for discrete
series.

Following their article, we fix a norm, denoted $\|\cdot\|$, on the Lie
algebra $\mathfrak{g}$ of $G$. We further define the function

$\textbf{r}:G\longrightarrow\mathbb{R}_{\geq
0},\quad\textbf{r}(g)=\textrm{log}\left(\textrm{max}\\{\|\textrm{Ad}(g)\|,\|\textrm{Ad}(g^{-1})\|\\}\right)$

so that, given $r\in\mathbb{R}_{>0}$, we can introduce the corresponding ball

$G_{<r}:=\\{g\in G|\textbf{r}(g)<r\\}.$

Given this set-up, we are in position to state their conjecture.

######  Conjecture 1.1 (Kazhdan-Yom Din, Asymptotic Schur Orthogonality
Relations).

Let $G$ be a semisimple group over a local field and let $(\pi,H)$ be a
tempered irreducible unitary representation of $G$. Then there exist
$\textbf{d}(\pi)\in\mathbb{Z}_{\geq 0}$ and
$\textbf{f}(\pi)\in\mathbb{R}_{>0}$ such that, for all
$v_{1},v_{2},v_{3},v_{4}\in H$, the following holds:

$\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}=\frac{1}{\textbf{f}(\pi)}\langle
v_{1},v_{3}\rangle\overline{\langle v_{2},v_{4}\rangle}.$

Assuming that the matrix coefficients involved are $K$-finite, one has the
following result:

######  Theorem 1.2 ([4], Theorem 1.7).

Let $(\pi,H)$ be a tempered irreducible unitary representation of $G$ and $K$
be a maximal compact subgroup of $G$. Then there exists
$\textbf{d}(\pi)\in\mathbb{Z}_{\geq 0}$ such that:

  1. (1)

If $G$ is non-Archimedean, there exists $\textbf{f}(\pi)\in\mathbb{R}_{>0}$
such that, for all $v_{1},v_{2},v_{3},v_{4}\in H_{K}$, we have

$\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}=\frac{1}{\textbf{f}(\pi)}\langle
v_{1},v_{3}\rangle\overline{\langle v_{2},v_{4}\rangle}.$

  2. (2)

If $G$ is Archimedean, for any given non-zero $v_{1},v_{2}\in H_{K}$, there
exists $C(v_{1},v_{2})>0$ such that

$\lim_{r\to\infty}\frac{\int_{G_{<r}}|\langle\pi(g)v_{1},v_{2}\rangle|^{2}\,dg}{r^{\textbf{d}(\pi)}}=C(v_{1},v_{2}).$

In the non-Archimedean case, the proof of (1) is achieved by first
establishing the validity of the analogous version of (2). The polarisation
identity allows the authors of [4] to define a form

$D(\cdot,\cdot,\cdot,\cdot):H_{K}\times H_{K}\times H_{K}\times
H_{K}\longrightarrow\mathbb{C}$

via the prescription

$D(v_{1},v_{2},v_{3},v_{4}):=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}.$

In [4], Section 4.1, this form is shown to be $G$-invariant and one would like
to invoke an appropriate form of Schur’s lemma to argue as in the standard
proof of Schur’s orthogonality relations. That is, for fixed $v_{2},v_{4}\in
H_{K}$, one defines the form

$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}:=D(\cdot,v_{2},\cdot,v_{4})$

and, for fixed $v_{1},v_{3}\in H_{K}$, the form

$\langle\cdot,\cdot\rangle^{v_{1},v_{3}}:=D(v_{1},\cdot,v_{3},\cdot).$

One applies Schur’s lemma to these forms, which implies that each such form is
a scalar multiple of the inner product on $H$. Upon comparing them, one
obtains the desired orthogonality relations.

The appropriate version of Schur’s lemma in the non-Archimedean case is
provided by Dixmier’s lemma, which can be applied since in the non-Archimedean
setting the subspace of K-finite vectors $H_{K}$ and the subspace of smooth
vectors $H^{\infty}$ coincide, the latter being equipped with the structure of
a Fréchet representation of $G$, which is irreducible since $H$ itself is
irreducible.

The purpose of this article is to prove that the analogue of (1) in Theorem
3.3 holds in the Archimedean case. As explained in [4], Section 4.2, it
suffices to prove the result for real semisimple groups (proved as Theorem
3.5).

######  Theorem 1.3.

Let $(\pi,H)$ be a tempered irreducible unitary representation of a real
semisimple group $G$. Let $K$ be a maximal compact subgroup of $G$. Then there
exists $\textbf{f}(\pi)\in\mathbb{R}_{>0}$ such that, for all
$v_{1},v_{2},v_{3},v_{4}\in H_{K}$, we have

$\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}=\frac{1}{\textbf{f}(\pi)}\langle
v_{1},v_{3}\rangle\overline{\langle v_{2},v_{4}\rangle}.$

We need to modify the strategy above to account for the fact that the spaces
of smooth and $K$-finite vectors of a Hilbert space representation $(\pi,H)$
of a real semisimple group do not coincide.

Our approach relies crucially on the admissibility of irreducible, unitary
representations of reductive groups, a foundational theorem proved by Harish
Chandra. The theory of admissible $(\mathfrak{g},K)$-modules then provides us
with the appropriate version of Schur’s lemma for $(\mathfrak{g},K)$-invariant
forms. Hence, we are reduced to verify that
$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}$ and
$\langle\cdot,\cdot\rangle^{v_{1},v_{3}}$ are, indeed,
$(\mathfrak{g},K)$-invariant. Having established this, to conclude the proof
of Theorem 1.3, we can argue as in [4], Section 4.

The article is organised as follows.

Section 2: We collect those notions and results on representation theory and
$(\mathfrak{g},K)$-modules that we need, including the admissibility of the
$(\mathfrak{g},K)$-modules associated to irreducible unitary representations,
and the appropriate form of Schur’s lemma.

Section 3: We establish the $(\mathfrak{g},K)$-invariance of the forms
introduced above and prove Theorem 1.3.

Acknowledgements: We would like to thank A. Yom Din for pointing out a problem
in an earlier draft of the article. The second author wishes to express his
deepest gratitude to Gaëtan Chenevier, to Antonella Perucca, and to his
advisor, Gabor Wiese, for their encouragement and advice.

##  2 Recollections on $(\mathfrak{g},K)$-modules

Our presentation of the theory of $(\mathfrak{g},K)$-modules, the core of this
section, follows [7] very closely. To discuss even its basic features, we need
to gather some results on unitary representations of compact groups. We begin
by recalling the basic notions in the study of representations of topological
groups, which we always assume to be Hausdorff.

First, following [7], Section 1.1, let $G$ denote a second-countable, locally
compact group, equipped with a left Haar measure $\,dg$, and let $V$ denote a
complex topological vector space. We denote by $\mathrm{GL}(V)$ the group of
invertible continuous endomorphisms of $V$. A representation of $G$ on $V$ is
a strongly continuous homomorphism $\pi:G\longrightarrow\mathrm{GL}(V)$. Let
$(\pi,V)$ denote the datum of a representation of $G$. A subspace of $V$ which
is stable under the action of $G$ through $\pi$ is called an invariant
subspace. A representation is said to be irreducible if the only closed
invariant subspaces are the trivial subspace and $V$ itself.

If $(H,\langle\cdot,\cdot\rangle)$ is a separable Hilbert space, a
representation $\pi$ of $G$ on $H$ is termed a Hilbert representation. If, in
addition, $G$ acts by unitary operators through $\pi$, the representation is
said to be unitary.

Next, following [6], Section 10, we introduce the basic features of the theory
of vector-valued integration.

Let $(X,\,dx)$ be a Radon measure space, let $H$ be a Hilbert space and assume
that

$f:X\longrightarrow H$

is measurable. The function $f$ is integrable if it satisfies the following
two conditions:

  1. (1)

For all $v\in H$,

$\int_{X}|\langle f(x),v\rangle|\,dx<\infty.$

  2. (2)

The map

$v\mapsto\int_{X}|\langle f(x),v\rangle|\,dx$

is a bounded conjugate-linear functional.

If $f:X\longrightarrow H$ is integrable, then, by the Riesz’ representation
theorem, there exists a unique element in $H$, denoted

$\int_{X}f(x)\,dx,$

such that, for all $v\in H$, we have

$\Bigl{\langle}\int_{X}f(x)\,dx,v\Bigr{\rangle}=\int_{X}\langle
f(x),v\rangle\,dx.$

######  Proposition 2.1.

Let $(X,\,dx)$ be as above. Let $H$, $E$ be Hilbert spaces,
$f:X\longrightarrow H$ a measurable function and $T:H\longrightarrow E$ a
bounded linear operator. Then the following holds:

  1. (1)

If

$\int_{X}\|f(x)\|\,dx<\infty,$

then $f:X\longrightarrow H$ is integrable.

  2. (2)

If $f:X\longrightarrow H$ is integrable, then so is $Tf:X\longrightarrow E$.
Moreover,

$T\left(\int_{X}f(x)\,dx\right)=\int_{X}Tf(x)\,dx.$

###### Proof.

See [6], Proposition 10.8 and Proposition 10.9. ∎

Now, let $(\pi,H)$ be a unitary representation of $G$. Let $v\in H$ and
$f:G\longrightarrow H$ be such that the map

$g\mapsto f(g)\pi(g)v$

is integrable. Let $\pi(f)v$ denote the unique element in $H$ such that, for
all $w\in H$, we have

$\langle\pi(f)v,w\rangle=\int_{G}f(g)\langle\pi(g)v,w\rangle\,dg.$

######  Proposition 2.2.

Let $(\pi,H)$ be as above. If $f\in L^{1}(G)$, then, for all $v\in H$, the map
$g\mapsto f(g)\pi(g)v$ is integrable and the prescription

$\pi(f):H\longrightarrow H,\text{ }v\mapsto\pi(f)v$

defines a bounded linear operator.

###### Proof.

See [6], Proposition 10.20. ∎

With the integral operators introduced in Proposition 2.2 at our disposal, we
have all the tools needed to state the main results on the unitary
representations of compact groups.

Let $K$ be a compact group. Let $\widehat{K}$ denote the set of equivalence
classes of irreducible unitary representations of $K$. If $(\pi,H)$ is a
unitary representation, for each $\gamma\in\widehat{K}$ let $H(\gamma)$ denote
the closure of the sum of all the closed invariant subspaces of $H$ in the
equivalence class of $\gamma$. We refer to $H(\gamma)$ as the
$\gamma$-isotypic component of $H$.

######  Proposition 2.3.

Let $K$ be a compact group. Let $(\pi,H)$ be an irreducible unitary
representation of $K$. Then $H$ is finite-dimensional.

###### Proof.

See [7], Proposition 1.4.2. ∎

Given Proposition 2.3, we can associate, to each $\gamma\in\widehat{K}$, the
function

$\chi_{\gamma}:K\longrightarrow\mathbb{C},\text{
}\chi_{\gamma}(g):=\mathrm{tr}\gamma(g),$

the character of $\gamma$.

Recall that if $\\{(\pi_{i},H_{i})|i\in I\\}$ is a countable family of unitary
representations of a topological group $G$, we can construct a new unitary
representation of $G$, the direct sum, on the Hilbert space completion of the
algebraic direct sum of the $H_{i}$’s. We refer the reader to [7], Section
1.4.1, for the details of this construction. We let

$\bigoplus_{i\in I}H_{i}$

denote the direct sum of the family $\\{(\pi_{i},H_{i})|i\in I\\}$, dropping
explicit reference to the $\pi_{i}$’s.

######  Proposition 2.4.

Let $K$ be a compact group. Let $(\pi,H)$ be a unitary representation of $K$.
Then $(\pi,H)$ is the direct sum representation of its $K$-isotypic
components; that is,

$H=\bigoplus_{\gamma\in\widehat{K}}H(\gamma).$

Moreover, let $\alpha_{\gamma}$ denote the function

$\alpha_{\gamma}(k):=\text{dim}(\gamma)\overline{\chi_{\gamma}}(k).$

Then the following holds:

$H(\gamma)=\pi(\alpha_{\gamma})H.$

###### Proof.

See [7], Lemma 1.4.7. ∎

######  Proposition 2.5.

Let $K$ be a compact group. If $(\pi,H)$ is a Hilbert space representation of
$K$, then there exists an inner product on $H$ that induces the original
topology on $H$ and for which $K$ acts unitarily through $\pi$.

###### Proof.

See [7], Lemma 1.4.8. ∎

We are finally ready to introduce $(\mathfrak{g},K)$-modules. Although they
can be defined for general Lie groups, as in [7], Section 3.3, we will assume
at the outset that our group $G$ is a reductive group. We adopt the definition
of reductive group that is given in [7], Chapter 2. Similarly for semisimple
groups.

######  Definition 2.6.

Let $G$ be a real reductive group with Lie algebra $\mathfrak{g}$. Let $K$ be
a maximal compact subgroup of $G$ with Lie algebra $\mathfrak{k}$. A vector
space $V$, equipped with the structure of $\mathfrak{g}$-module and
$K$-module, is called a $(\mathfrak{g},K)$-module if the following conditions
hold:

  1. (1)

For all $v\in V$, for all $X\in\mathfrak{g}$, for all $k\in K$,

$kXv=Ad(k)Xkv$

  2. (2)

For all $v\in V$, the span of the set

$Kv:=\\{kv|k\in K\\}$

is a finite-dimensional subspace of $V$, on which the action of $K$ is
continuous.

  3. (3)

For all $v\in V$, for all $Y\in\mathfrak{k}$,

$\frac{d}{dt}\mathrm{exp}(tY)v|_{t=0}=Yv.$

We remark that (3) implicitly uses the smoothness of the action of $K$ on the
span of $Kv$. This follows from the fact that a continuous group homomorphism
between Lie groups is automatically smooth.

Let $V$ and $W$ be $(\mathfrak{g},K)$-modules and let
$\text{Hom}_{\mathfrak{g},K}(V,W)$ denote the space of
$\mathfrak{g}$-morphisms that are also $K$-equivariant. Then $V$ and $W$ are
said to be equivalent if $\text{Hom}_{\mathfrak{g},K}(V,W)$ contains an
invertible element.

A $(\mathfrak{g},K)$-module $V$ is called irreducible if the only subspaces
that are invariant under the actions of $\mathfrak{g}$ and $K$ are the trivial
subspace and $V$ itself. In this case, we have the following theorem:

######  Theorem 2.7.

Let $V$ be an irreducible $(\mathfrak{g},K)$-module. Then the space
$\text{Hom}_{\mathfrak{g},K}(V,V)$ is $1$-dimensional.

###### Proof.

This is the result actually proved in [7], Lemma 3.3.2, although the statement
there says $\text{Hom}_{\mathfrak{g},K}(V,W)$, for an unspecified $W$. We
believe it is a typo. ∎

Let $V$ be a $(\mathfrak{g},K)$-module. Since, given each $v\in V$, the span
of $Kv$, say $W_{v}$, is a finite-dimensional continuous representation of
$K$, we can use Proposition 2.5 and then apply Proposition 2.4, thus
decomposing $W_{v}$ into a finite sum of finite-dimensional $K$-invariant
subspaces of $V$. For $\gamma\in\widehat{K}$, we let $V(\gamma)$ denote the
sum of all the $K$-invariant finite dimensional subspaces in the equivalence
class of $\gamma$. Then the discussion above implies that

$V=\bigoplus_{\gamma\in\widehat{K}}V(\gamma)$

as a $K$-module, with the direct sum indicating the algebraic direct sum. A
$(\mathfrak{g},K)$-module $V$ is called admissible if, for all
$\gamma\in\widehat{K}$, $V(\gamma)$ is finite-dimensional.

Given a unitary representation $(\pi,H)$, there exists a
$(\mathfrak{g},K)$-module naturally associated to it. To define it, recall
that a vector $v\in H$ is called smooth if the map

$g\mapsto\pi(g)v$

is smooth. Let $H^{\infty}$ denote the subspace of smooth vectors of $H$. It
is a standard fact that the prescription

$\dot{\pi}(X):=\frac{d}{dt}\pi(\mathrm{exp}(tX))v|_{t=0},$

for $v\in H^{\infty}$ and $X\in\mathfrak{g}$, defines an action of
$\mathfrak{g}$ on $H^{\infty}$. Recall that a vector $v\in H$ is $K$-finite if
the span of the set

$\pi(K)v:=\\{\pi(k)v|k\in K\\}$

is finite-dimensional. Let $H_{K}$ denote the subspace of $K$-finite vectors
of $H$. By [7], Lemma 3.3.5, with the action of $\mathfrak{g}$ so defined and
with the action of $K$ through $\pi$, the space $H_{K}\cap H^{\infty}$ is a
$(\mathfrak{g},K)$-module. The representation $(\pi,H)$ is said to be
admissible if $H_{K}\cap H^{\infty}$ is admissible as a
$(\mathfrak{g},K)$-module and $(\pi,H)$ is called infinitesimally irreducible
if $H_{K}\cap H^{\infty}$ is irreducible as a $(\mathfrak{g},K)$-module. It is
in general not true that a $K$-finite vector is smooth. However, if $(\pi,H)$
is admissible, we have the following result:

######  Theorem 2.8.

Let $G$ be a real reductive group and $(\pi,H)$ an admissible representation
of $G$. Then every $K$-finite vector is smooth.

###### Proof.

See [8], Theorem 4, part 1. ∎

In light of the following fundamental result of Harish Chandra, Theorem 2.8
will play an important role in this article.

######  Theorem 2.9.

Let $(\pi,H)$ be an irreducible unitary representation of $G$. Then $(\pi,H)$
is admissible.

###### Proof.

See [7], Theorem 3.4.10. ∎

As in [7], Section 3.3.6, given a $(\mathfrak{g},K)$-module $V$, we can equip
its algebraic dual $V^{*}$ with the following actions of $K$ and
$\mathfrak{g}$. For all $k\in K$, for all $\mu\in V^{*}$, we define

$\widetilde{k}\mu(v):=\mu(k^{-1}v),$

for all $v\in V$. For all $X\in\mathfrak{g}$, for all $\mu\in V^{*}$, we
define

$\widetilde{X}\mu(v):=-\mu(Xv),$

for all $v\in V$. The subspace of $V^{*}$ consisting of the $K$-finite
elements with respect to the action of $K$ so defined is a
$(\mathfrak{g},K)$-module: the dual, or contragredient, $\widetilde{V}$, of
$V$. Similarly, we can equip the space of conjugate-linear functionals on $V$
with analogously defined actions. If $\mu$ is a conjugate-linear functional on
$V$, for all $k\in K$ and for all $v\in V$, we define

$\overline{k}\mu(v):=\mu(k^{-1}v).$

For all $X\in\mathfrak{g}$ and for all $v\in V$, we define

$\overline{X}\mu(v):=-\mu(Xv).$

The subspace of $K$-finite conjugate-linear functionals on $V$, equipped with
these actions, is a $(\mathfrak{g},K)$-module: the conjugate-dual,
$\overline{V}$, of $V$.

Both the following statements are extracted from [7]. The first appears at the
end of the proof of Lemma 4.3.2, while the second is essentially given in
Lemma 4.3.3. Since we shall need them in the third section, we decided to fill
in the details of the proofs. The first part of the argument in the next
proposition is an adaption of [3], Proposition 8.5.7. The key idea is to
exploit the local finiteness of $(\mathfrak{g},K)$-modules to reduce to a
setting where the projection to a given $K$-isotypic component of an arbitrary
element is given by an appropriate integral operator, as in Theorem 2.4.

######  Proposition 2.10.

Let $V$ be an admissible $(\mathfrak{g},K)$-module. Then

$\widetilde{V}=\bigoplus_{\gamma\in\widehat{K}}(V(\gamma)^{*}).$

In particular, $\widetilde{V}$ is admissible.

###### Proof.

Let $\mu\in\widetilde{V}$, let $W$ be defined as

$W:=\text{Span}\\{\widetilde{K}\mu\\}.$

Then $W$ is a finite-dimensional representation of $K$, which, by Proposition
2.5, we can assume unitary. By decomposing $\mu$ according to its $K$-isotypic
components, we can reduce to the case that $\mu\in W(\gamma)$ and, by
Proposition 2.4, that

$\mu=\widetilde{k}(\alpha_{\widetilde{\gamma}})\mu^{\prime},$

for some $\mu^{\prime}\in W$. Here,
$\widetilde{k}(\alpha_{\widetilde{\gamma}})\mu^{\prime}$ denotes the
functional obtained as the $W$-valued integral of the function

$k\mapsto\alpha_{\widetilde{\gamma}}(k)\widetilde{k}\mu^{\prime}.$

We want to show that $\mu$ vanishes identically outside $V(\gamma)$. So, let
$v\in V$ and let $E$ be defined as

$E:=\text{Span}\\{Kv\\}.$

Then, as above, $E$ is a finite-dimensional unitary representation of $K$ and
we have

$E(\gamma)=k(\alpha_{\gamma})E$

We can compute as follows:

$\begin{split}\mu(v)&=\widetilde{k}({\alpha_{\widetilde{\gamma}}})\mu^{\prime}(v)\\\
&=\int_{K}\alpha_{\widetilde{\gamma}}(k)\widetilde{k}\mu^{\prime}(v)\,dk\\\
&=\int_{K}\alpha_{\widetilde{\gamma}}(k)\mu^{\prime}(k^{-1}v)\,dk\\\
&=\mu^{\prime}\left(\int_{K}\alpha_{\widetilde{\gamma}}(k)k^{-1}v\,dk\right)\\\
&=\mu^{\prime}\left(\int_{K}\alpha_{\gamma}(k^{-1})k^{-1}v\,dk\right)\\\
&=\mu^{\prime}(k(\alpha_{\gamma})v).\end{split}$

Here, the second equality is justified by a reasoning that we have learnt from
P. Garrett: we view evaluation at $v\in V$ as a linear functional on $W$.
Since $W$ is finite-dimensional, this functional is continuous and we can
interchange it with the integral

$\int_{K}\alpha_{\widetilde{\gamma}}(k)\widetilde{k}\mu^{\prime}\,dk,$

which is $W$-valued. The fourth equality follows from the fact that
$\mu^{\prime}$ is continuous on the finite-dimensional vector space $E$, hence
we can interchange it with an integral valued in $E$. The fifth equality
follows from

$\chi_{\widetilde{\gamma}}(k)=\chi_{\gamma}(k^{-1}),$

a fact proved in [2], Chapter II, Proposition 4.10, (vi). Finally, we used the
unimodularity of $K$. Hence, we showed that $\mu\in\widetilde{V}(\gamma)$ acts
on $v\in V$ through the projection of $v$ to its $\gamma$-isotypic component.
This implies that $\mu$ vanishes identically outside $V(\gamma)$ and gives an
element in $V(\gamma)^{*}$. To complete the proof, it is enough to observe
that we can decompose any element in $\widetilde{V}$ uniquely according to its
$K$-isotypic components and apply the discussion above to each resulting
summand.

For the reverse inclusion, Let $\mu$ be an element of the right-hand side of
the display in the statement. Then there exist
$\mu_{\gamma_{1}},\dots,\mu_{\gamma_{n}}$, with $\mu_{\gamma_{i}}\in
V(\gamma_{i})^{*}$ for each $i$, such that

$\mu=\sum_{i=1}^{n}\mu_{\gamma_{i}}.$

Admissibility of $V$ implies that $V(\gamma)$ is finite-dimensional for every
$\gamma\in\widehat{K}$, hence so is $V(\gamma)^{*}$. Since
$\widetilde{K}\mu_{\gamma_{i}}$ is contained in $V(\gamma_{i})^{*}$, it
follows that, for each $i$, the span of $\widetilde{K}\mu_{\gamma_{i}}$ is
finite-dimensional. Hence, the same holds for the span of $\widetilde{K}\mu$,
proving that the right-hand side of the above display is contained in
$\widetilde{V}$. ∎

######  Proposition 2.11.

Let $(\pi,H)$ be an irreducible unitary representation. For $w\in H_{K}$, let
$\phi_{w}$ denote the linear functional on $H_{K}$ defined by
$\phi_{w}(v)=\langle v,w\rangle$, and set

$\Phi(H_{K}):=\\{\phi_{w}|w\in H_{K}\\}.$

Then the following equality holds:

$\widetilde{H_{K}}=\Phi(H_{K}).$

###### Proof.

By Theorem 2.9 and by Proposition 2.10, we have

$\widetilde{H_{K}}=\bigoplus_{\gamma\in\widehat{K}}(H_{K}(\gamma)^{*})$

with $H_{K}(\gamma)$ finite-dimensional for each $\gamma\in\widehat{K}$.
Hence, each $H_{K}(\gamma)$ inherits a Hilbert space structure from $H$. Let
$\mu\in\widetilde{H_{K}}$ and let $\mu_{\gamma_{1}},\dots,\mu_{\gamma_{n}}$,
with $\mu_{\gamma_{i}}\in H_{K}(\gamma_{i})^{*}$, be such that

$\mu=\sum_{i=1}^{n}\mu_{\gamma_{i}}.$

For each $i$, by Riesz representation theorem, we can find $w_{i}\in H_{K}$
such that, for all $v\in H_{K}$,

$\mu_{\gamma_{i}}(v)=\langle v,w_{i}\rangle.$

Then, setting

$w:=\sum_{i=1}^{n}w_{i}\in H_{K},$

we obtain, for all $v\in H_{K}$,

$\mu(v)=\langle v,w\rangle,$

proving that $\widetilde{H_{K}}$ is a subset of $\Phi(H_{K})$.

For the reverse inclusion, let $\phi_{w}\in\Phi(H_{K})$. Let
$w_{1},\dots,w_{m}\in H_{K}$ be a basis for the space spanned by $\pi(K)w$
and, for $k\in K$, let $\lambda_{1},\dots,\lambda_{m}\in\mathbb{C}$ be such
that

$\pi(k)w=\sum_{i=1}^{m}\lambda_{i}w_{i}.$

Since

$\widetilde{k}\phi_{w}(v)=\langle\pi(k)^{-1}v,w\rangle=\langle
v,\pi(k)w\rangle=\sum_{i=1}^{m}\overline{\lambda_{i}}\phi_{w_{i}}(v),$

it follows that the space spanned by $\widetilde{K}\phi_{w}$ is contained in
the subspace spanned by $\phi_{w_{1}},\dots,\phi_{w_{m}}$. Hence,
$\phi_{w}\in\widetilde{H_{K}}$, concluding the proof. ∎

We conclude this quick (and far from complete) review of the theory of
$(\mathfrak{g},K)$-modules by proving the version of Schur’s lemma for
sesquilinear forms that we will use in Section 3. This is given by Corollary
2.14 below. First, we need:

######  Theorem 2.12.

Let $(\pi,H)$ be an admissible representation. Then $(\pi,H)$ is irreducible
if and only if it is infinitesimally irreducible.

###### Proof.

See [7], Theorem 3.4.11. ∎

######  Theorem 2.13.

Let $V$ be an admissible $(\mathfrak{g},K)$-module. Suppose that there exist a
$(\mathfrak{g},K)$-module $W$ and a non-degenerate
$(\mathfrak{g},K)$-invariant sesquilinear form

$B(\cdot,\cdot):V\times W\longrightarrow\mathbb{C}.$

Then $W$ is $(\mathfrak{g},K)$-isomorphic to $\overline{V}$.

###### Proof.

This is [7], Lemma 4.5.1, except for the fact that our form is sesquilinear.
To account for it, we modify the definition of the map $T$ in the reference by
setting, for a given $w\in W$, $T(w)(v)=B(w,v)$ for all $v\in V$. This defines
a map from $W$ to $\overline{V}$ obtained by sending $w$ to $T(w)$ which, by
the argument in the reference, is a $(\mathfrak{g},K)$-isomorphism. ∎

The next corollary is proved by adapting to our case the argument in [3],
Proposition 8.5.12 and using the beginning of the proof of [5], Proposition
9.1.

######  Corollary 2.14.

Let $(\pi,H)$ be an irreducible admissible representation. Then, up to a
constant, there exists at most one non-zero $\mathfrak{(}g,K)$-invariant
sesquilinear form on $H_{K}$. In particular, if $(\pi,H)$ is irreducible
unitary, then every such form is a constant multiple of
$\langle\cdot,\cdot\rangle$.

###### Proof.

The irreducibility of $H$ implies that of $H_{K}$, by Theorem 2.12 and by
Theorem 2.8. Let $B(\cdot,\cdot)$ be a $(\mathfrak{g},K)$-invariant
sesquilinear form. Consider the linear subspace $V_{0}$ of $H_{K}$ defined as

$V_{0}:=\\{v\in H_{K}|B(v,w)=0\text{ for all }w\in H_{K}\\}.$

Since $B(\cdot,\cdot)$ is non-zero, $V_{0}$ is a proper subspace of $H_{K}$.
Since $B(\cdot,\cdot)$ is moreover $(\mathfrak{g},K)$-invariant, it follows
that $V_{0}$ is a $(\mathfrak{g},K)$-invariant subspace of $H_{K}$, hence, by
the irreducibility of $H_{K}$, it must be zero. Analogous considerations for
the subspace

$V^{0}:=\\{w\in H_{K}|B(v,w)=0\text{ for all }v\in H_{K}\\}$

imply that $B(\cdot,\cdot)$ is non-degenerate. By Theorem 2.13, the map
$v\mapsto T(v)$, $T(v)(\cdot):=B(v,\cdot)$, is a
$(\mathfrak{g},K)$-isomorphism. Since $H_{K}$ is irreducible, the space
$\text{Hom}_{\mathfrak{g},K}(H_{K},H_{K})$ is $1$-dimensional by Theorem 2.7.
Now, let $B^{\prime}(\cdot,\cdot)$ be another such form, with associated
isomorphism $T^{\prime}$. Then $T(T^{\prime})^{-1}=cI$, for some
$c\in\mathbb{C}$. For the last statement, the unitarity of $(\pi,H)$ implies
that $\langle\cdot,\cdot\rangle$ is a $(\mathfrak{g},K)$-invariant non-
degenerate sesquilinear form and Theorem 2.9, with the discussion above,
implies the result. ∎

##  3 Schur orthogonality

We fix the datum of a real semisimple group $G$ and a maximal compact subgroup
$K$ of $G$.

For a fixed $d\in\mathbb{Z}_{\geq 0}$, let $\mathcal{L}_{d}$ be the set of
measurable functions $f:G\longrightarrow\mathbb{C}$ such that

$\lim_{r\to\infty}\frac{\int_{G_{<r}}|f(g)|^{2}\,dg}{r^{d}}<\infty.$

Let $\|\cdot\|_{d}$ denote the following prescription:

$\|f(g)\|_{d}:=\left(\lim_{r\to\infty}\frac{\int_{G_{<r}}|f(g)|^{2}\,dg}{r^{d}}\right)^{\frac{1}{2}}.$

Recall that for all $x\in G$, we can define the right and left action of $G$
on a function $f$ as $R(x)f(g):=f(gx)$ and $L(x)f(g):=f(x^{-1}g)$,
respectively.

######  Proposition 3.1.

Let $f\in\mathcal{L}_{d}$. Then the following holds:

  1. (1)

Let $x\in G$ be such that both $R(x)f$ and $R(x^{-1})f$ belong to
$\mathcal{L}_{d}$. Then $\|R(x)f\|_{d}=\|f\|_{d}$. The analogous statement is
true for $L(x)f$.

  2. (2)

$\|f(g^{-1})\|_{d}=\|f(g)\|_{d}$

###### Proof.

For (1), we observe that, for a given $x\in G$, the right translate
$(G_{<r})x$ of $G_{<r}$ is contained in $G_{<r+\textbf{r}(x)}$. Indeed, by
definition, $g\in G_{<r}$ implies $\textrm{log}\|\textrm{Ad}(g)\|<r$ and
$\textrm{log}\|\textrm{Ad}(g^{-1})\|<r$. Thus,
$\textrm{log}\|\textrm{Ad}(gx)\|<r+\textrm{log}\|\textrm{Ad}(x)\|$ and
$\textrm{log}\|\textrm{Ad}((gx)^{-1})\|<r+\textrm{log}\|\textrm{Ad}(x^{-1})\|$,
which implies

$\begin{split}\textbf{r}(gx)&<r+\textrm{max}\\{\textrm{log}\|\textrm{Ad}(x)\|,\textrm{log}\|\textrm{Ad}(x^{-1})\|\\}\\\
&=r+\textrm{log}\left(\textrm{max}\\{\|\textrm{Ad}(x)\|,\|\textrm{Ad}(x^{-1})\|\\}\right)\\\
&=r+\textbf{r}(x).\end{split}$

Now, for $f$ an element in $\mathcal{L}_{d}$ as in the statement, we compute
as follows:

$\begin{split}\|R(x)f(g)\|_{d}^{2}&=\|f(gx)\|_{d}^{2}\\\
&=\lim_{r\to\infty}\frac{\int_{G_{<r}}|f(gx)|^{2}\,dg}{r^{d}}\\\
&=\lim_{r\to\infty}\frac{\int_{G}|f(gx)|^{2}\mathds{1}_{G_{<r}}(g)\,dg}{r^{d}}\\\
&=\lim_{r\to\infty}\frac{\int_{G}|f(g)|^{2}\mathds{1}_{G_{<r}}(gx^{-1})\,dg}{r^{d}}\\\
&=\lim_{r\to\infty}\frac{\int_{(G_{<r})x}|f(g)|^{2}\,dg}{r^{d}}\\\
&\leq\lim_{r\to\infty}\frac{\int_{G_{<r+\textbf{r}(x)}}|f(g)|^{2}\,dg}{r^{d}}\\\
&=\|f(g)\|^{2}_{d}.\end{split}$

where, in the fourth equality, we used the unimodularity of $G$.
We remark that, without the assumption on $R(x)f$, we could only prove the
weaker inequality

$\limsup_{r\to\infty}\frac{\int_{G_{<r+\textbf{r}(x)}}|f(g)|^{2}\,dg}{r^{d}}\leq\|f(g)\|_{d}.$

Coming back to our proof, we obtained

$\|R(x)f\|_{d}\leq\|f\|_{d}.$

Similarly, we have

$\|R(x^{-1})f\|_{d}\leq\|f\|_{d}.$

In this last inequality, substituting $R(x)f$ with $f$, we obtain
$\|f\|_{d}\leq\|R(x)f\|_{d}$, which implies
$\|f\|_{d}\leq\|R(x)f\|_{d}\leq\|f\|_{d}$. The argument for $L(x)f$ runs
analogously. For (2), it suffices to observe that $(G_{<r})^{-1}=G_{<r}$ and
to use unimodularity. ∎

The set $\mathcal{L}_{d}$ is not a vector space and $\|\cdot\|_{d}$ is not a
seminorm on it. As A. Yom Din pointed out, following [1], one could remedy the
issue by using a limit superior in the definition of $\|\cdot\|_{d}$. However,
we will not need anything but Proposition 3.1 in this article.

For a tempered irreducible unitary representation $(\pi,H)$ of $G$, for given
$v,w\in H$, let

$\phi_{v,w}(g):=\langle\pi(g)v,w\rangle$

denote the associated matrix coefficient. By (2) of Theorem 1.2, there exists
$d(\pi)\in\mathbb{Z}_{\geq 0}$ such that $\phi_{v,w}\in\mathcal{L}_{d(\pi)}$
for all $v,w\in H_{K}$.

As in Section 4.1 of [4], by the polarisation identity and by (2) of Theorem
1.2, the prescription

$D(v_{1},v_{2},v_{3},v_{4}):=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}$

is a well-defined form on $H_{K}$ that is linear in the first and fourth
variable, conjugate-linear in the second and the third.

As explained in the introduction, we consider the following family of forms.
For fixed $v_{2},v_{4}\in H_{K}$, we define

$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}:=D(\cdot,v_{2},\cdot,v_{4}),$

which is linear in the first variable and conjugate linear in the second. For
fixed $v_{1},v_{3}\in H_{K}$, we define

$\langle\cdot,\cdot\rangle^{v_{1},v_{3}}:=D(v_{1},\cdot,v_{3},\cdot),$

which is conjugate-linear in the first variable and linear in the second.

As in [4], Section 4.1, showing that for all $k_{1},k_{2},k_{3},k_{4}\in K$
and for all $v_{2},v_{2},v_{3},v_{4}\in H_{K}$ we have

$D(\pi(k_{1})v_{1},\pi(k_{2})v_{2},\pi(k_{3})v_{3},\pi(k_{4})v_{4})=D(v_{1},v_{2},v_{3},v_{4}),$

can be reduced, by the polarisation identity, to prove that, with the notation
of (2) of Theorem 1.2, we have

$C(\pi(k_{1})v,\pi(k_{2})w)=C(v,w)$

for all $k_{1},k_{2}\in K$ and for all $v,w\in H_{K}$. This will imply the
$K$-invariance of $\langle\cdot,\cdot\rangle^{v_{1},v_{3}}$ and
$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}$.

######  Proposition 3.2.

Let $(\pi,H)$ be a tempered irreducible unitary representation of a real
semisimple group $G$. Then, for all $k_{1},k_{2}\in K$, for all $v,w\in
H_{K}$, we have

$\lim_{r\to\infty}\frac{\int_{G_{<r}}|\langle\pi(g)\pi(k_{1})v,\pi(k_{2})w\rangle|^{2}\,dg}{r^{\textbf{d}(\pi)}}=\lim_{r\to\infty}\frac{\int_{G_{<r}}|\langle\pi(g)v,w\rangle|^{2}\,dg}{r^{\textbf{d}(\pi)}}.$

Hence, for all $k\in K$, and for all $v_{1},v_{2},v_{3},v_{4}\in H_{K}$, we
have

$\langle\pi(k)v_{1},\pi(k)v_{3}\rangle_{v_{2},v_{4}}=\langle
v_{1},v_{3}\rangle_{v_{2},v_{4}}$

and

$\langle\pi(k)v_{2},\pi(k)v_{4}\rangle^{v_{1},v_{3}}=\langle
v_{2},v_{4}\rangle^{v_{1},v_{3}}.$

###### Proof.

We observe that

$\langle\pi(g)\pi(k_{1})v,\pi(k_{2})w\rangle=\phi_{v,w}(k_{2}^{-1}gk_{1})=L(k_{2})R(k_{1})\phi_{v,w}(g).$

Since $\pi(k_{1})v\in H_{K}$, by (2) of Theorem 1.2 it follows that
$R(k_{1})\phi_{v,w}\in\mathcal{L}_{d(\pi)}$. Therefore, we also have
$L(k_{2})R(k_{1})\phi_{v,w}\in\mathcal{L}_{d(\pi)}$. Using (1) of Proposition
3.1, we compute

$\begin{split}\lim_{r\to\infty}\frac{\int_{G_{<r}}|\langle\pi(g)\pi(k_{1})v,\pi(k_{2})w\rangle|^{2}\,dg}{r^{\textbf{d}(\pi)}}&=\|L(k_{2})R(k_{1})\phi_{v,w}(g)\|_{\textbf{d}(\pi)}^{2}\\\
&=\|\phi_{v,w}(g)\|_{\textbf{d}(\pi)}^{2},\end{split}$

as required. The $K$-invariance of the forms
$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}$ and
$\langle\cdot,\cdot\rangle^{v_{1},v_{3}}$ now follows from the polarisation
identity. ∎

Next, we proceed to prove that the forms we consider are
$\mathfrak{g}$-invariant. First, we observe that if $v_{2},v_{3},v_{4}\in
H_{K}$, the linear functional

$\langle\cdot,v_{3}\rangle_{v_{2},v_{4}}:H_{K}\longrightarrow\mathbb{C}$

is $K$-finite, hence it belongs to the contragredient
$(\mathfrak{g},K)$-module, $\widetilde{H_{K}}$, of $H_{K}$.

######  Lemma 3.3.

Let $v_{2},v_{3},v_{4}\in H_{K}$. Then the linear functional

$\langle\cdot,v_{3}\rangle_{v_{2},v_{4}}:H_{K}\longrightarrow\mathbb{C}$

is $K$-finite; that is, it belongs to $\widetilde{H_{K}}$.

###### Proof.

Since $v_{3}\in H_{K}$, the span of $\pi(K)v_{3}$ is finite-dimensional and we
let $w_{1},\dots,w_{n}\in H_{K}$ be a basis for it. For $k\in K$, let
$\lambda_{1},\dots,\lambda_{n}\in\mathbb{C}$ be such that

$\pi(k)v_{3}=\sum_{i=1}^{n}\lambda_{i}w_{i}.$

We compute

$\begin{split}\widetilde{k}\langle
v,v_{3}\rangle_{v_{2},v_{4}}&=\langle\pi(k)^{-1}v,v_{3}\rangle_{v_{2},v_{4}}\\\
&=\langle v,\pi(k)v_{3}\rangle_{v_{2},v_{4}}\\\
&=\sum_{i=1}^{n}\overline{\lambda_{i}}\langle
v,w_{i}\rangle_{v_{2},v_{4}},\end{split}$

where the second equality follows by $K$-invariance of
$\langle\cdot,\cdot\rangle_{v_{2},v_{4}}$.
Therefore, the span of $\widetilde{K}\langle\cdot,v_{3}\rangle_{v_{2},v_{4}}$
is contained in the span of the functionals
$\langle\cdot,w_{1}\rangle_{v_{2},v_{4}},\dots,\langle\cdot,w_{n}\rangle_{v_{2},v_{4}}$.
∎

######  Proposition 3.4.

Let $(\pi,H)$ be a tempered irreducible unitary representation of a real
semisimple group $G$. Then, for all $X\in\mathfrak{g}$, and for all
$v_{1},v_{2},v_{3},v_{4}\in H_{K}$, we have

$\langle\dot{\pi}(X)v_{1},v_{3}\rangle_{v_{2},v_{4}}=-\langle
v_{1},\dot{\pi}(X)v_{3}\rangle_{v_{2},v_{4}}$

and

$\langle\dot{\pi}(X)v_{2},v_{4}\rangle^{v_{1},v_{3}}=-\langle
v_{2},\dot{\pi}(X)v_{4}\rangle^{v_{1},v_{3}},$

where $\dot{\pi}$ denotes the action of $\mathfrak{g}$ through $\pi$
introduced in Section 2, page 8.

###### Proof.

Since the functional $\langle\cdot,v_{3}\rangle_{v_{2},v_{4}}$ belongs to
$\widetilde{H_{K}}$ by Lemma 3.3 and since $H_{K}$ is admissible by Theorem
2.9, it follows from Proposition 2.11 that there exists $w\in H_{K}$ such that

$\langle v,v_{3}\rangle_{v_{2},v_{4}}=\langle v,w\rangle$

for all $v\in H_{K}$. For $X\in\mathfrak{g}$, we compute

$\begin{split}\langle\dot{\pi}(X)v_{1},v_{3}\rangle_{v_{2},v_{4}}&=\langle\dot{\pi}(X)v_{1},w\rangle\\\
&=\frac{d}{dt}\langle\pi(\mathrm{exp}(tX))v_{1},w\rangle|_{t=0}\\\
&=\frac{d}{dt}\langle v_{1},\pi(\mathrm{exp}(-tX))w\rangle|_{t=0}\\\
&=\frac{d}{dt}\overline{\langle\pi(\mathrm{exp}(-tX))w,v_{1}\rangle}|_{t=0}\\\
&=\overline{\langle-\dot{\pi}(X)w,v_{1}\rangle}\\\
&=\overline{X}\overline{\langle w,v_{1}\rangle}\\\
&=\overline{X}\overline{\langle v_{3},v_{1}\rangle}_{v_{4},v_{2}}\\\
&=-\overline{\langle\dot{\pi}(X)v_{3},v_{1}\rangle}_{v_{4},v_{2}}\\\
&=-\langle v_{1},\dot{\pi}(X)v_{3}\rangle_{v_{2},v_{4}}\end{split}$

Here, the notation $\overline{X}$ denotes the action of $\mathfrak{g}$ on the
conjugate-dual $(\mathfrak{g},K)$-module $\overline{H_{K}}$, introduced in
Section 2, page 9. The sixth equality follows from viewing
$\overline{\langle\cdot,v_{1}\rangle}$ as an element in $\overline{H_{K}}$,
the conjugate-dual of $H_{K}$ (See Section 2, after Theorem 2.9). For the
seventh equality, we observe that

$\langle v_{1},w\rangle=\langle v_{1},v_{3}\rangle_{v_{2},v_{4}}$

implies that

$\langle w,v_{1}\rangle=\overline{\langle v_{1},w\rangle}=\overline{\langle
v_{1},v_{3}\rangle}_{v_{2},v_{4}}=\langle v_{3},v_{1}\rangle_{v_{4},v_{2}}.$

Finally, the eighth equality follows from viewing
$\overline{\langle\cdot,v_{1}\rangle}_{v_{4},v_{2}}$ as an element in
$\overline{H_{K}}$. For the $\mathfrak{g}$-invariance of the form
$\langle\cdot,\cdot\rangle_{v_{1},v_{3}}$, the computation

$\begin{split}\langle
v_{2},v_{4}\rangle^{v_{1},v_{3}}&=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}\\\
&=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle
v_{1},\pi(g^{-1})v_{2}\rangle\overline{\langle
v_{3},\pi(g^{-1})v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}\\\
&=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle
v_{1},\pi(g)v_{2}\rangle\overline{\langle
v_{3},\pi(g)v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}\\\
&=\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{4},v_{3}\rangle\overline{\langle\pi(g)v_{2},v_{1}\rangle}\,dg}{r^{\textbf{d}(\pi)}}\\\
&=\langle v_{4},v_{2}\rangle_{v_{3},v_{1}},\end{split}$

where the third equality follows from (2) of Proposition 3.1, reduces the
statement to the previous case. ∎

We can now complete the strategy outlined in the introduction.

######  Theorem 3.5.

Let $(\pi,H)$ be a tempered irreducible unitary representation of a real
semisimple group $G$ and let $K$ be a maximal compact subgroup of $G$. Then
there exists $\textbf{f}(\pi)\in\mathbb{R}_{>0}$ such that, for all
$v_{1},v_{2},v_{3},v_{4}\in H_{K}$, we have

$\lim_{r\to\infty}\frac{\int_{G_{<r}}\langle\pi(g)v_{1},v_{2}\rangle\overline{\langle\pi(g)v_{3},v_{4}\rangle}\,dg}{r^{\textbf{d}(\pi)}}=\frac{1}{\textbf{f}(\pi)}\langle
v_{1},v_{3}\rangle\overline{\langle v_{2},v_{4}\rangle}.$

###### Proof.

Fix $v_{2},v_{4}\in H_{K}$. By Proposition 3.4, we can apply Corollary 2.14 to
the form $\langle\cdot,\cdot\rangle_{v_{2},v_{4}}$. Hence there exists
$c_{v_{2},v_{4}}\in\mathbb{C}$ such that for all $v_{1},v_{3}\in H_{K}$ we
have

$\langle v_{1},v_{3}\rangle_{v_{2},v_{4}}=c_{v_{2},v_{4}}\langle
v_{1},v_{3}\rangle.$

Similarly, fixing $v_{1},v_{3}\in H_{K}$, there exists a
$d_{v_{1},v_{3}}\in\mathbb{C}$ such that

$\overline{\langle v_{4},v_{2}\rangle}^{v_{3},v_{1}}=d_{v_{1},v_{3}}\langle
v_{4},v_{2}\rangle,$

since the left-hand side is conjugate-linear in the first variable. Hence,
since

$\overline{\langle v_{4},v_{2}\rangle}^{v_{3},v_{1}}=\langle
v_{2},v_{4}\rangle^{v_{1},v_{3}},$

we obtain

$\langle v_{2},v_{4}\rangle^{v_{1},v_{3}}=d_{v_{1},v_{3}}\overline{\langle
v_{2},v_{4}\rangle}$

By definition, we have

$D(v_{1},v_{2},v_{3},v_{4})=\langle v_{1},v_{3}\rangle_{v_{2},v_{4}}=\langle
v_{2},v_{4}\rangle^{v_{1},v_{3}},$

so, for a vector $v_{0}\in H_{K}$ of norm $1$, using (2) of Theorem 1.2, we
obtain a real number $C(v_{0},v_{0})>0$ such that

$D(v_{0},v_{0},v_{0},v_{0})=C(v_{0},v_{0})=c_{v_{0},v_{0}}=d_{v_{0},v_{0}}.$

Computing $D(v_{1},v_{0},v_{3},v_{0})$, we have

$d_{v_{1},v_{3}}=c_{v_{0},v_{0}}\langle v_{1},v_{3}\rangle.$

Therefore, we obtained

$D(v_{1},v_{2},v_{3},v_{4})=c_{v_{0},v_{0}}\langle
v_{1},v_{3}\rangle\overline{\langle v_{2},v_{4}\rangle},$

showing that $\textbf{f}(\pi):=\frac{1}{C(v_{0},v_{0})}$ does not depend on
the choice of $v_{0}$, as required. ∎

## References

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(Anne-Marie Aubert) Sorbonne Université and Université Paris Cité, CNRS, IMJ-
PRG, F-75005 Paris, France.

Email address<EMAIL_ADDRESS>

(Alfio Fabio La Rosa) Department of Mathematics, University of Luxembourg,
Maison du Nombre, 6 Avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg.

Email address<EMAIL_ADDRESS>