# End-to-end optimized image compression for multiple machine tasks

Lahiru D. Chamain†∗ Fabien Racapé† Jean Bégaint† Akshay Pushparaja† Simon
Feltman†
†InterDigital - AI Lab ∗University of California, Davis 4410 El Camino Real 1
Shields Ave Los Altos, CA, 94022, USA Davis, CA, 95616, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>Work carried out
while Lahiru D. Chamain was interning at the InterDigital AI Lab.

###### Abstract

An increasing share of captured images and videos are transmitted for storage
and remote analysis by computer vision algorithms, rather than to be viewed by
humans. Contrary to traditional standard codecs with engineered tools, neural
network based codecs can be trained end-to-end to optimally compress images
with respect to a target rate and any given differentiable performance metric.
Although it is possible to train such compression tools to achieve better
rate-accuracy performance for a particular computer vision task, it could be
practical and relevant to re-use the compressed bit-stream for multiple
machine tasks. For this purpose, we introduce ‘Connectors’ that are inserted
between the decoder and the task algorithms to enable a direct transformation
of the compressed content, which was previously optimized for a specific task,
to multiple other machine tasks. We demonstrate the effectiveness of the
proposed method by achieving significant rate-accuracy performance improvement
for both image classification and object segmentation, using the same bit-
stream, originally optimized for object detection.

###### Abstract

In this document, we provide additional details on the experiments we carried
out to help reproduce the results presented in the paper. Furthermore,
additional sample images reconstructed from COCO, ImageNet-1K and Cityscapes
data sets are shown to demonstrate the effectiveness of the proposed
connectors.

###### Index Terms:

Image compression, multiple machine tasks

###### Index Terms:

Image compression, multiple machine tasks

##  I Introduction

Many computer vision tasks, such as object detection, tracking, image
classification, etc., rely on computationally challenging algorithms that
cannot be run on embedded devices, and require the content to be transmitted
to remote servers. Hence, the emerging research field of ‘Video Coding for
Machines’ (VCM) aims to develop optimized compression methods for video
content that is meant to be analyzed by machines, rather than viewed by human
users [1].

When the image/video content is analyzed by machines (e.g. pedestrian and
vehicle detection by smart cars), the codec and the considered task (we call
it ‘primary task’) algorithm can be optimized end-to-end to achieve the
optimal task accuracy at a given rate [2]. In this paper, we study the case
where the optimization of the codec for the primary task does not consider the
fidelity of the decoded frames, usually measured using Peak Signal-to-Noise
Ratio (PSNR). Contrary to traditional compression standards such as JPEG [3],
or H.265/HEVC [4], decoded images are not optimized to ensure high fidelity or
viewing quality, as can be seen in Fig. 1. However the decoded content is
still viewable [5], which is useful for applications that involve human
supervision. For instance, a video surveillance system may require the
produced videos to be used as pieces of evidence.

The features extracted from Convolutional Neural Networks (CNN) are the
primary candidates for most visual tasks [6]. However, the codecs that are
optimized end-to-end for a primary task cannot be directly used on off-the-
shelf secondary tasks models trained on RGB images even though they can
produce features that are useful for visual machine tasks. How to adapt these
features directly for visual machine tasks without piling up in complexity is
still unanswered.

In this paper, we discuss how to adapt the features selected by a codec
optimized for a primary task for secondary vision tasks performed on different
data sets.

Rate-Distortion | Rate-Detection | Rate-Distortion | Rate-Multi task
---|---|---|---
|  |  |
Acc mAP: 32.6 |  34.1 ($+$4.6%) | Acc top-5: 66.43% |  75.18% ($+$8.75%)
PSNR: 30.49 dB | 16.31 dB | 26.37 dB |  22.35 dB ($-$4.02)
Figure 1: End-to-end optimization of an ANN-based codec for multiple machine
tasks. Left: the detection accuracy is improved by 4.6% at the same bitrate by
optimizing the codec from [7] for rate-detection instead of the conventional
rate-distortion based on PSNR (‘bear’ image from COCO [8]). Right: We show
that it is possible to use a rate-‘object detection’ optimized codec to
achieve high rate-‘multi task accuracy’ performances (classification,
segmentation, etc.), using the proposed simple pre-trained connectors. ‘Tiger’
from ImageNet-1K [9] is compressed using the same rate-‘object detection’
optimized codec coupled with a trained connector (81 parameters) to achieve a
8.75% image classification accuracy improvement on an off-the-shelf classifier
(right image).

In particular we propose simple, one-time trained and task-independent
‘Connectors’ that transform primary-task-optimized-features during off-the-
shelf inference for secondary visual tasks. Further we show that fine tuning
the secondary task models on primary task optimized features leads to
significant gain in rate-accuracy performances compared to baseline models.
These observed gains result from the richness of the primary task data set,
essential features learned by end-to-end optimization of the codec-primary
task at a given bandwidth and the feature transformation of the task
independent connector at the inference. See Fig. 1. Finally, based on the
visual tasks such as object detection, classification and segmentation, we
demonstrate the effectiveness of the proposed connectors with interesting
visual examples. Additional visual examples and details of the experiments are
given in the supplemental material.

##  II Related Works

In terms of optimizing image/video compression codecs for a given learning
task, only a handful of works emerges from the literature. The authors of [10]
and [2] proposed to learn the quantization tables of JPEG and JPEG2000,
respectively, using end-to-end training for image classification with
differential codec implementations. Using a learned entropy coder, the authors
of [11] has shown the effectiveness of end-to-end optimizing features for
image classification. Similarly in [5], the authors end-to-end optimize the
RGB images compressed with scale hyperpriors codec [7] for object detection on
COCO data set and show significant rate-detection performance gains at lower
bandwidths. In this work, we utilize the rate-detection optimized codec in [5]
as the primary task optimized codec.

In the previous works that explore for multi-task applications, we observe
that tasks reusing the same resources are carefully selected to be closely
related and beneficial to each other [12]. In order to extract generic
descriptors, in [6] authors use ImageNet data set [9] as it consists of over
1.2 million images and the task as fine grained classification.

In [13], authors assured the possibility of reusing the features among visual
tasks through transfer learning without added complexity. By defining a
structure connecting several visual tasks, they identify the redundancy
between tasks. Furthermore the authors of [10], discussed how the learned
features generalize over different architectures for the same learning task by
observing that the codec learned for the MobileNet [14] classifier achieves
similar gains with a ResNet [15] classifier.

##  III Compression for multiple machine task

In order to find the features that are vital for common visual machine tasks
such as detection, classification and segmentation etc., we propose the
following strategy. First we select the important features that are non-
redundant based on a single (primary) task $T$ and then transform the selected
features at the inference for other (secondary) tasks $S$.

###  III-A Structure of networked codec and task algorithm

The considered networked Machine Learning (ML) inference applications observe
the general structure shown in Fig. 2. It consists of a ‘codec’ that
efficiently compresses the input $\mathbf{x}$ and two ‘task models’ that
perform the primary task $T$ and the secondary task $S$ on the reconstructed
images $\mathbf{\hat{x}}_{\beta}^{J}$. We parameterize the end-to-end
(jointly) optimized codec that encodes images at a quality parameterized by
$\beta$ with $\boldsymbol{\psi}_{\beta}^{J}$,
($\mathbf{\hat{x}}_{\beta}^{J}=\boldsymbol{\psi}_{\beta}^{J}(\mathbf{x})$) and
the models for tasks $T$ and $S$ with $\boldsymbol{\theta}$ and
$\boldsymbol{\phi}$. The pre-trained connector $C$ parameterized by
$\boldsymbol{c}_{\beta}$ applied only for the secondary task $S$ at the
inference.

In general, the codec consists of an Encoder Analyzer (E), an Entropy Coder
(EC), an Entropy Decoder (ED) and a Decoder Synthesizer (D). See [5] for more
details. During inference for task $T$, a pre-trained task model
$\boldsymbol{\theta}$ performs the learning on $\mathbf{\hat{x}}_{\beta}^{J}$
and estimates the result $y^{T}$,
($\boldsymbol{\theta}(\mathbf{\hat{x}}_{\beta}^{J})=y^{T}$). In practice, the
task model $\boldsymbol{\phi}$ is optimized for minimum task loss on the input
images of best possible (reference) quality ‘$rq$’ available during training.
We denote the task model optimized for the reference quality $rq$ by
$\boldsymbol{\phi}_{rq}$.

###  III-B Primary task selection

Since the purpose of the primary task is to select general features for visual
machine tasks, following [6], we select a challenging and more general task
for this. Ex: Object detection with COCO data set [8]. This involves
separating the image patches with objects of interest with different scales
and orientations and then classifying them in to classes. With this choice of
task and data set compared to the work [6], we expect to extract more generic
features by using the bounding box information along with class labels.

Figure 2: General structure of networked ML use case: input $\mathbf{x}$ is
encoded to latent $\mathbf{z}$ and stored/transmitted over a channel as a
binary stream. The decoder decodes the binary stream and reconstructs
$\mathbf{\hat{x}}_{\beta}^{J}$ on which task $T$.is performed. Connector $C$
is applied before the inference for task $S$.

###  III-C Task specific optimization with Joint fine tuning: J-FT

The goal here is to fully optimize the system for the primary task. Hence, we
choose to jointly optimize the codec parameters $\boldsymbol{\psi}$ and task
parameters $\boldsymbol{\theta}$ to converge at the best rate-task accuracy
performance. Keeping the structure of the decoder which outputs RGB images
enables us to directly use off-the-shelf task models without redesigning the
task models to take latent features as inputs.

Assume that $\mathcal{X}_{rq}$ denotes the set containing reference quality
images $\mathbf{x}\in\mathcal{X}_{rq}$ and $y_{gt}^{T}\in\mathcal{Y}^{T}$ is
the ground truth for the inputs $\mathbf{x}$ for the considered task $T$. We
express the loss function for a task $T$ with
$\mathcal{L}_{T}(y^{T},y_{gt}^{T})$ which calculates task loss based on the
prediction $y^{T}$ generated by the task model $\boldsymbol{\theta}$ for the
input and the ground truth pair $(\mathbf{\hat{x}},y_{gt}^{T})$ for task $T$.

Hence, we can find the optimal codec-task model parameters combination
($\boldsymbol{\psi}_{\beta}^{J},\boldsymbol{\theta}_{\beta}^{J}$) by
minimizing the task loss and the rate [5, 2, 11].

$\displaystyle\boldsymbol{\psi}_{\beta}^{J},\boldsymbol{\theta}_{\beta}^{J}$
$\displaystyle=\arg\min_{\boldsymbol{\psi},\boldsymbol{\theta}}\dfrac{1}{|\mathcal{X}_{rq}|}\sum_{\mathbf{x},y_{gt}^{T}\in\mathcal{X}_{rq},\mathcal{Y}^{T}}\\{\mathcal{L}_{T}(\boldsymbol{\theta}(\boldsymbol{\psi}(\mathbf{x})),y_{gt})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\beta\mathcal{L}_{R}(\mathbf{z})\\}$
(1)

$\mathcal{L}_{R}(\mathbf{z})$ denotes the entropy $H(\mathbf{z})$ of the
latent representation $\mathbf{z}$. See Fig 2. The control parameter $\beta$
regulates the trade-off between the accuracy of task $T$ and the
bandwidth/rate of the input to the task model.

###  III-D Introducing Connectors

Even though the optimally selected features for a given bandwidth with the
primary task are important for a secondary task $S$, the scale of the features
are not fine tuned for the pre-trained models of task $S$ that expects images
of reference quality $\mathbf{\hat{x}}_{rq}$ as inputs. Hence, the end-to-end
optimized inputs $\mathbf{\hat{x}}_{\beta}^{J}$ perform poorly in terms of
rate-accuracy performance for task $S$ on a pre-trained off-the-shelf task
model $\boldsymbol{\phi}_{rq}$ trained with rate-distortion optimized inputs.

In order to de-specialize the optimized images $\mathbf{\hat{x}}_{\beta}^{J}$,
we add a connector $C$ parameterized by $\boldsymbol{c}$ in between the codec
output $\mathbf{\hat{x}}_{\beta}^{J}$ and the task model
$\boldsymbol{\phi}_{rq}$ to obtain $\mathbf{\hat{x}}^{C}_{\beta}$ which is the
input to the model $\boldsymbol{\phi}_{rq}$. In the context of re-usability of
features that are originally optimized for a primary task, on off-the-shelf
secondary task models, the codec $\boldsymbol{\psi}_{\beta}^{J}$ and the pre-
trained model $\boldsymbol{\phi}_{rq}$ are fixed, hence beyond out control. We
explore the possibility of a simple fixed/trainable connector that can be
stored at the end devices that perform inference for multiple tasks.

For trainable $C$ we propose to find the optimal parameter setting
$\boldsymbol{c}_{\beta}^{J}$ that minimizes the visual distortion
$\mathcal{L}_{D}$ between the connector produced image
$\mathbf{\hat{x}}_{\beta}^{C}$ and its rate-distortion optimized version
$\mathbf{x}_{rq}^{D}$. Therefore we find $\boldsymbol{c}_{\beta}^{J}$ s.t.

$\displaystyle\boldsymbol{c}_{\beta}^{J}=\arg\min_{\boldsymbol{c}}\dfrac{1}{|\mathcal{X}_{rq}|}\sum_{\mathbf{x}\in\mathcal{X}_{rq}}\mathcal{L}_{D}(\boldsymbol{c}(\boldsymbol{\psi}_{\beta}^{J}(\mathbf{x})),\mathbf{x}).$
(2)

For each rate-accuracy control parameter $\beta$, the trained connector
$\boldsymbol{c}_{\beta}^{J}$ can be stored at the decoder and used for images
that were compressed with the codec $\boldsymbol{\psi}_{\beta}^{J}$ during
inference of any secondary task according to the following flow for the input,
ground truth pair ($\mathbf{x}$, $y_{gt}^{S}\in\mathcal{Y}^{S}$). Here $y^{S}$
is the estimated label for task $S$.

$\mathbf{x}\xrightarrow[\boldsymbol{\psi}_{\beta}^{J}]{\text{E-EC-
ED-D}}\mathbf{\hat{x}}_{\beta}^{J}\xrightarrow[\boldsymbol{c}_{\beta}^{J}]{C}\mathbf{\hat{x}}^{C}_{\beta}\xrightarrow[\boldsymbol{\phi}_{rq}]{S}y$
(3)

Note that the proposed optimization of $\boldsymbol{c}$ is independent of the
secondary task $S$. Therefore training of the connector given in Eq. (2) is
only performed once and zero knowledge of the secondary task is required.

###  III-E Fine-tuning the task model with end-to-end optimized images: J-FT
T-FT

Given that we have the access to the secondary task model and training set,
for better accuracy at task $S$, we can fine tune the task model
$\boldsymbol{\phi}$ on images compressed and reconstructed using the end-to-
end jointly optimized coded $\boldsymbol{\psi}_{\beta}^{J}$.

The fine-tuned task model $\boldsymbol{\phi}_{\beta}$ is obtained by
minimizing the task loss $\mathcal{L}_{S}$ as the following at each rate
specified by $\beta$.

$\displaystyle\boldsymbol{\phi}_{\beta}=\arg\min_{\boldsymbol{\phi}}\dfrac{1}{|\mathcal{X}_{rq}|}\sum_{\mathbf{x},y_{gt}^{S}\in\mathcal{X}_{rq},\mathcal{Y}^{S}}\mathcal{L}_{S}(\boldsymbol{\phi}(\boldsymbol{\psi}_{\beta}^{J}(\mathbf{x})),y_{gt}^{S})$
(4)

##  IV Experiments and Results

As the primary task $T$, we used the well researched object detection on
COCO-2017 [8] data set. COCO-2017 dataset for object detection contains over
180k training and 5k validation RGB images of average size over 256$\times$256
with annotated bounding boxes belonging to 80 classes. We used the standard
primary challenge metric ‘box mAP’ to evaluate the detection accuracy of the
val set and the model Faster-RCNN [16] with ResNet-50 backbone following [5]
to reproduce their results shown as J-FT in [5].

Similarly as the image codec for this paper, we used Scale hyperpriors model
bmshj2018-hyperprior, optimized on MSE [7] which is an auto encoder-decoder
based differentiable codec featuring a learned entropy coder.

###  IV-A End-to-end codec optimization for the primary task

Following Eq. (III-C), we fine tuned the codec and the detector for 6 epochs
with the same optimizers and learning rates used in [5] and regenerated the
results (shown in Fig. 3 of [5]). Note that for joint end-to-end training, the
rate-detection performance gain is significant compared to codec only (C-FT)
and task only (T-FT) fine tunning at lower bit rates.

###  IV-B Application of Connectors

As the secondary tasks for inference, we selected image classification
($S_{1}$) on ImageNet-1k [9] and semantic segmentation ($S_{1}$) on Cityscapes
[17]. ImageNet-1K consists of RGB images of size over 256$\times$256 belonging
to 1000 classes, each of them containing about 1300 training and 50 validation
samples. We evaluated the performance of the classification task using top-1
and top-5 accuracy as metrics. Cityscapes is a video dataset recorded in
streets from 50 cities captured in 5000 frames split into 2975 training, 500
validation and 1525 test samples. For our experiments we used the left frames
of training and validation sets with fine annotations belonging to 19 classes.
The task of per-pixel semantic labeling is to label each pixel into one of 19
classes measured by the metric IoU [17].

We explore the performance of 3 simple blocks as candidates for the connecor:
1) C-Avgpool: 2D average pooling block of size 3$\times$3 (ablation) 2)
C-DepthConv: 2D depth-wise convolution block 3$\times$3 (27 parameters) 3)
C-Conv: 2D convolution block of size 3$\times$3 (81 parameters). To train the
connectors C-deptConv and C-Conv, we used COCO-2017 validation set as ground
truths. According to Eq. (2), we minimized the MSE (as $\mathcal{L}_{D}$)
between the ground truths and the connector output images using the Adam
optimizer with an initial learning rate of 0.1 reduced by a factor of 0.1
after each 10 epochs for a total of 30 epochs.

(a) ImageNet-1K

(b) Cityscapes

Figure 3: Inference results on the validation sets with connectors. baseline:
distortion optimized images, J-FT: images compressed and decompressed by the
codec optimized for detection, C-Avgpool, C-DepthConv, C-Conv: J-FT images
after each connector

Fig. 3(a) shows the top-1 accuracy on the ImageNet-1K validation set with off-
the-shelf ResNet-50 classifier. Interestingly, the features optimized for
COCO-detection achieve better classification accuracy compared to the images
generated by rate-distortion optimized codecs on off-the-shelf ResNet trained
with rate-distortion optimized images. Image classification being part of
COCO-detection task can be a reason for this behavior. Note that all 3
connectors achieve similar gains for ImageNet-1K inference which is
significantly higher compared to the extrapolated inference on rate-distortion
optimized images (baseline) at lower data rates.

Similarly Fig. 3(b) shows the results for semantic segmentation on Cityscapes
data set. For inference, we used a Deeplabv3+ [18] with ResNet-50 backbone
trained from scratch on the training set of the ‘left 8 bit’ original images
with fine annotations as the ground truths. Using the trained model, we
repeated the similar inference steps on Cityscapes val set with the connectors
trained with COCO val set.

The same pre-trained connectors are used on ImageNet-1K and Cityscapes.
Contrary to ImageNet-1K results, C-Conv connector shows significant gains for
the segmentation task on Cityscapes. Overall, the pre-trained connector
enables the reuse of decoded images (optimized for a primary task) for
secondary tasks even with better rate-accuracy performance.

###  IV-C Effect of connectors on visualization

Original |  J-FT[5] | C-Avgpool | C-Conv
---|---|---|---
|  |  |
PSNR | 17.93 | 21.34 | 21.86
BPP | 0.0687 | 0.0687 | 0.0687
|  |  |
PSNR | 20.73 | 25.03 | 27.36
BPP | 0.2298 | 0.2298 | 0.2298
Figure 4: Comparison of the visual quality of an image from Cityscapes data
set. Columns J-FT [5]: Rate-detection optimized image. C-Avgpool, C-Conv:
output of each connector.

Since Eq. (III-C) does not include any criterion related to the visual quality
of decoded images, or distortion constraints with respect to the original, the
codec optimized for object detection (J-FT [5]) produces low fidelity images.
Fig. 4 shows the reconstructed images from the output of C-Avgpool and C-Conv.
It is clear that J-FT produces images that include high frequency features,
which help detecting objects. The proposed connectors generally smoothen the
details of decoded images, specific to the primary task, making them more
generic for any secondary task or viewing. Since the proposed training of the
connector aim to minimize the MSE between rate-detection optimized and rate-
distortion optimized images, training of the connector once per given rate
(BPP) is sufficient.

In both cases, connectors output higher quality compared to J-FT [5] both
visually and quantitatively (PSNR). Further examples and details including
PSNR and MS-SSIM curves on the COCO dataset, are provided in supplement.

###  IV-D Fine-tuning the classifier for better performance

Being able to train/fine-tune the secondary task model, we observe that the
latent/feature maps optimized for a primary task (detection) achieves
significant accuracy gains for the secondary tasks from fine-tuning as well
compared to fine-tuning on rate-distortion optimized images.

(a) ImageNet-1K

(b) Citysapes

Figure 5: Fine-tuning the classifier. T-FT: fine-tuned task on distortion
optimized decoded images. J-FT: pre-trained task on rate-detection optimized
images. J-FT T-FT: fine-tuned task on rate-detection optimized images.

To demonstrate the validity of this claim, we fine-tuned the secondary task
models with rate-detection optimized images separately at each rate point
(J-FT T-FT). As a baseline to compare with, we fine-tuned the secondary task
models (T-FT) on images compressed with rate-distortion optimized
bmshj2018-hyperprior. Fig. 5(a) and 5(b) compares the accuracy gains for
ImageNet-1K and Cityscapes respectively.

We fine-tuned the classifier using an SGD optimizer minimizing the
classification cross entropy at a learning rate of 0.001 and a momentum of 0.9
for 5 epochs for both bmshj2018-hyperprior T-FT and J-FT T-FT settings. For
Cityscapes segmentation, we used an SGD optimizer for 50 epochs, with a
momentum of 0.9 and a polynomial learning rate decay strategy initialized at
0.01 with a decay of 5e-4.

For instance, the proposed method achieves 70% top-1 classification accuracy
at 0.21 BPP which corresponds to 55% bandwidth saving compared to rate-
distortion optimized bmshj2018-hyperprior T-FT. The J-FT T-FT setting records
3% classification accuracy improvement at 0.2 BPP

##  V Conclusion

In this paper, we introduced simple pre-trained connectors, adapting the
images decoded from a compressed bit-stream originally optimized for object
detection to other secondary vision tasks, such as image classification and
segmentation. Based on experiments on ImageNet-1K and Cityscapes data sets, we
showed that the proposed connectors achieve significantly better rate-accuracy
performance compared to the conventional codecs optimized for rate-distortion.
In future work, we plan to explore the applicability of those connectors to
video compression for multiple machine tasks.

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##  VI Rate-Detection Baseline

We experimented with Scale hyperprior image compression code originally
optimized for Mean Square Error (MSE).

###  VI-A Scale hyperprior

We used the implementation of bmshj2018-hyperprior method from [7], of which a
pytorch version is available at https://github.com/InterDigitalInc/CompressAI
with MSE-optimized pre-trained models. Fig. 6 shows the basic building blocks
of the scale hyperprior codec. For any input image $\mathbf{x}$, the Encoder
analysis $g_{a}$ produces a quantized tensor $\mathbf{z}_{1}$, containing the
mean values of the coefficients of the latent, and their standard deviations
(known as scale hyperpriors) $\mathbf{z}_{2}$. The tensors $\mathbf{z}_{1}$
and $\mathbf{z}_{2}$ are encoded using arithmetic coding and the bit stream is
transmitted or stored. At the destination, $g_{s}$ synthesizes the
reconstructed image $\mathbf{\hat{x}}$ from the arithmetically decoded
$\mathbf{z}_{1}$ and $\mathbf{z}_{2}$. The total Rate of the bit stream for
each image is the sum of the Entropies $H(\mathbf{z}_{1})$ and
$H(\mathbf{z}_{2})$.

Figure 6: Structure of Scale hyperprior codec

The blocks $g_{a}$, $h_{a}$, $g_{s}$ and $h_{s}$ consist of 2d-convolutional
layers and generalized divisive normalization (GDN/IGDN) layers. For the codec
optimization with a fixed decoder setting (J-FT-FD), we only updated the
parameters in $g_{a}$ and $h_{a}$during training.

###  VI-B Faster-RCNN model

As the detection model, we used the off-the shelf Faster-RCNN with ResNet-50
[15] backbone available at https://github.com/open-mmlab/mmdetection. In order
to achieve better detection performance, COCO-2017 validation images are
resized (scaled-up) to the resolution $800\times 1200$ at the pre-processing
stage before feeding to the detector.

##  VII Visual artifacts on J-FT optimized images

In Fig. 7 we compare the visual quality of the images generated with the rate-
distortion optimized Scale-hyperprior codec $\psi_{q}^{D}$ as described in [5]
and the end-to-end rate-detection optimized Scale-hyperprior codec
$\psi_{\beta}^{J}$. Images decoded from the end-to-end method optimized for
object detection clearly show visible artifacts, which highlights the features
used for detection. These artifacts result from the optimization based on
compression rate and detection accuracy, using a loss function that does not
account for the (visual) distortion.

$\psi_{q}^{D}$ |  |  |  |
---|---|---|---|---
| $q=4$ | $q=3$ | $q=2$ | $q=1$
BPP | 0.2875 | 0.2054 | 0.1461 | 0.1024
$\psi_{\beta}^{J}$ |  |  |  |
| $\beta=0.1$ | $\beta=0.3186$ | $\beta=0.6675$ | $\beta=1.0$
BPP | 0.2472 | 0.1453 | 0.0943 | 0.0751
$\psi_{q}^{J\rm{-FT-FD}}$ |  |  |  |
| $\beta=0.1$ | $\beta=0.3186$ | $\beta=0.6675$ | $\beta=1.0$
BPP | 0.2818 | 0.1662 | 0.1069 | 0.0813
Figure 7: Visible artifacts on end-to-end optimized images.

In addition to visible artifacts, the images generated with the end-to-end
optimized codec with a fixed decoder show color distortions, since the decoder
has been optimized for rate-distortion.

##  VIII Feature reuse with connectors

In this section we present some additional results that help benchmark the
performance of the proposed connectors. Fig. 8 illustrates the top-1 and top-5
ImageNet-1K [9] classification accuracy on the validation set. Note that both
metrics top-1 and top-5 show similar improvements at the rates $\leq$ 0.225
BPP.

(a) top-1

(b) top-5

Figure 8: Inference on ImageNet-1K. baseline: distortion optimized images,
J-FT : images compressed and decompressed with codec optimized for detection
(J-FT), C-Avgpool: J-FT images after average pooling, C-DepthConv: J-FT images
after 2D depth-wise conv. connector, C-Conv: J-FT w/o C images after 2D conv.

Similarly, Fig. 9 gives the top-1 and top-5 ImageNet-1K classification
accuracy on the validation set when fine tuning a ResNet-50 classifier. We
observe that fine-tuning the classifier with the images that are end-to-end
optimized for detection facilitates the classification, compared to rate-
distortion optimized images.

(a) top-1

(b) top-5

Figure 9: Fine-tuning the classifier on rate-detection optimized ImageNet-1K.
baseline: inference T-FT: fine-tuning on distortion optimized images. J-FT :
inference on rate-detection optimized images J-FT T-FT: fine-tuning on rate-
detection optimized images.

Even though rate-detection-optimized images show better results for secondary
tasks, the visual quality of the reconstructed images is significantly lower,
compared with rate-distortion or rate-accuracy-distortion[10] optimized
images, as displayed in Fig. 7. The numerical measures of distortion in PSNR
and MS-SSIM can be found in Fig. 10. Note that the gain of PSNR from the
proposed connectors is more significant, compared to MS-SSIM, since the
connectors are optimized to minimize the mean square error (MSE).

(a) PSNR

(b) MS-SSIM

Figure 10: Rate-distortion characteristics comparison of the proposed methods.

We selected the object detection task on COCO-2017 as the primary task
considering its high complexity and the variability of the data set. By doing
so, we expected the end-to-end optimized codec to capture a fair amount of
prior information that is common to vision based learning tasks. Furthermore
with the intention of preserving the features that is backbone dependent (if
there is any), we used models with ResNet-50 backbone for all 3 tasks in the
experiments.

##  IX Comparison of the reconstruction quality of sample images from
ImageNet-1K and Cityscapes

In this section we present additional reconstructed images to highlight the
effects of the proposed connectors. The reported rate values for each sample
correspond to the average BPP value for the validation set containing that
sample image.

Figure 11: Reconstruction quality for ImageNet-1K Original | J-FT | J-FT Avg pool | J-FT Conv
---|---|---|---
|  |  |
PSNR | 15.55 | 17.36 | 18.10
BPP | 0.1119 | 0.1119 | 0.1119
|  |  |
PSNR | 18.19 | 20.67 | 21.14
BPP | 0.2203 | 0.2203 | 0.2203
Figure 12: Reconstruction quality for ImageNet-1K Original | J-FT | J-FT Avg pool | J-FT Conv
---|---|---|---
|  |  |
PSNR | 15.55 | 17.36 | 18.10
BPP | 0.1119 | 0.1119 | 0.1119
|  |  |
PSNR | 18.19 | 20.67 | 21.14
BPP | 0.2203 | 0.2203 | 0.2203
Figure 13: Reconstruction quality for ImageNet-1K Original | J-FT | J-FT Avg pool | J-FT Conv
---|---|---|---
|  |  |
PSNR | 15.20 | 17.11 | 17.30
BPP | 0.1119 | 0.1119 | 0.1119
|  |  |
PSNR | 16.51 | 18.02 | 19.54
BPP | 0.2203 | 0.2203 | 0.2203
Figure 14: Reconstruction quality for Cityscapes Original | J-FT | J-FT Avg pool | J-FT Conv
---|---|---|---
|  |  |
PSNR | 17.93 | 21.34 | 21.86
BPP | 0.0687 | 0.0687 | 0.0687
|  |  |
PSNR | 20.73 | 25.03 | 27.36
BPP | 0.2298 | 0.2298 | 0.2298

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# Invariant and hyperinvariant subspaces for amenable operators

Luo Yi Shi  Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>,  YU Jing Wu  Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>and  You Qing Ji  Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA<EMAIL_ADDRESS>
(Date: 7.04. 2010)

# Invariant and hyperinvariant subspaces for amenable operators

Luo Yi Shi  Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>,  YU Jing Wu  Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>and  You Qing Ji  Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA<EMAIL_ADDRESS>
(Date: 7.04. 2010)

###### Abstract.

There has been a long-standing conjecture in Banach algebra that every
amenable operator is similar to a normal operator. In this paper, we study the
structure of amenable operators on Hilbert spaces. At first, we show that the
conjecture is equivalent to every non-scalar amenable operator has a non-
trivial hyperinvariant subspace and equivalent to every amenable operator is
similar to a reducible operator and has a non-trivial invariant subspace; and
then, we give two decompositions for amenable operators, which supporting the
conjecture.

###### Key words and phrases:

Amenable; Invariant subspaces; Hyperinvariant subspaces; Reduction property

###### 2000 Mathematics Subject Classification:

47C05 (46H35 47A65 47A66 47B15)

Supported by NCET(040296), NNSF of China(10971079) and the Specialized
Research Fund for the Doctoral Program of Higher Education(20050183002)

##  1\. Introduction

Throughout this paper, $\mathfrak{H}$ denotes a complex separable infinite-
dimension Hilbert space and $\mathfrak{B}(\mathfrak{H})$ denotes the bounded
linear operators on $\mathfrak{H}$. For an algebra $\mathfrak{A}$ in
$\mathfrak{B}(\mathfrak{H})$, we write $\mathfrak{A}^{\prime}$ for the
commutant of $\mathfrak{A}$ (i.e.,
$\mathfrak{A}^{\prime}=\\{B\in\mathfrak{B}(\mathfrak{H}),BA=AB~{}~{}\textup{for
all}A\in\mathfrak{A}\\})$ and $\mathfrak{A}^{\prime\prime}$ for the double
commutant of $\mathfrak{A}$ (i.e.,
$\mathfrak{A}^{\prime\prime}=(\mathfrak{A}^{\prime})^{\prime}$). We also write
Lat$\mathfrak{A}$ for the collection of those subspaces which are invariant
for every operator in $\mathfrak{A}$. We say $\mathfrak{A}$ is completely
reducible if for every subspace $M$ in Lat$\mathfrak{A}$ there exists $N$ in
Lat$\mathfrak{A}$ such that $\mathfrak{H}=M\dot{+}N$ (i.e., $M\cap N=\\{0\\}$
and $\mathfrak{H}$ is the algebraic direct sum of $M$ and $N$); $\mathfrak{A}$
is reducible if for every subspace $M$ in Lat$\mathfrak{A}$ we have $M^{\bot}$
(the orthogonal complement of $M$) in Lat$\mathfrak{A}$; $\mathfrak{A}$ is
transitive if Lat$\mathfrak{A}=\\{\\{0\\},\mathfrak{H}\\}$. If
$T\in\mathfrak{B}(\mathfrak{H})$, we say that a subspace $M$ of $\mathfrak{H}$
is a hyperinvariant subspace for $T$ if $M$ is invariant under each operator
in $\mathfrak{A}_{T}^{\prime}$; $M$ is a reducible subspace for $T$ if
$M,M^{\bot}\in$ Lat$T$.

The concept of amenable Banach algebras was first introduced by B. E. Johnson
in [10]. Suppose that $\mathfrak{A}$ is a Banach algebra. A Banach
$\mathfrak{A}$-bimodule is a Banach space $X$ that is also an algebra
$\mathfrak{A}$-bimodule for which there exists a constant $K>0$ such that
$||a\cdot x||\leq K||a||||x||$ and $||x\cdot a||\leq K||a||||x||$ for all
$a\in\mathfrak{A}$ and $x\in X$. We note that $X^{*}$, the dual of $X$, is a
Banach $\mathfrak{A}$-bimodule with respect to the dual actions

$[a\cdot f](x)=f(x\cdot a),[f\cdot a](x)=f(a\cdot x),a\in\mathfrak{A},x\in
X,f\in X^{*}.$

Such a Banach $\mathfrak{A}$-bimodule is called a dual
$\mathfrak{A}$-bimodule.

A derivation $D:\mathfrak{A}\rightarrow X$ is a continuous linear map such
that $D(ab)=a\cdot D(b)+D(a)\cdot b$, for all $a,b\in\mathfrak{A}$. Given
$x\in X$, the inner derivation $\delta_{x}:\mathfrak{A}\rightarrow X$, is
defined by $\delta_{x}(a)=a\cdot x-x\cdot a$.

According to Johnson s original definition, a Banach algebra $\mathfrak{A}$ is
amenable if every derivation from $\mathfrak{A}$ into the dual
$\mathfrak{A}$-bimodule $X^{*}$ is inner for all Banach
$\mathfrak{A}$-bimodules $X$. If $T\in\mathfrak{B}(\mathfrak{H})$, denote the
norm-closure of span$\\{T^{k}:k\in\\{0\\}\cup\mathbb{N}\\}$ by
$\mathfrak{A}_{T}$, where $\mathbb{N}$ is the set of natural numbers, $T$ is
said to be an amenable operator, if $\mathfrak{A}_{T}$ is an amenable Banach
algebra. Ever since its introduction, the concept of amenability has played an
important role in research in Banach algebras, operator algebras and harmonic
analysis. There has been a long-standing conjecture in the Banach algebra
community, stated as follows:

######  Conjecture 1.1.

A commutative Banach subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable if
and only if it is similar to a $C^{*}$-algebra.

One of the first result in this direction is due to Willis [16]. Willis showed
that if $T$ is an amenable compact operator, then $T$ is similar to a normal
operator. In [7] Gifford studied the reduction property for operator algebras
consisting of compact operators and showed that if such an algebra is amenable
then it is similar to a $C^{*}$-algebras. In the recent papers [5], [6]
Farenick, Forrest and Marcoux showed that if $T$ is similar to a normal
operator, then $\mathfrak{A}_{T}$ is amenable if and only if
$\mathfrak{A}_{T}$ is similar to a $C^{*}$-algebra and the spectrum of $T$ has
connected complement and empty interior; If $T$ is a triangular operator with
respect to an orthonormal basis of $\mathfrak{H}$, then $\mathfrak{A}_{T}$ is
amenable if and only if $T$ is similar to a normal operator whose spectrum has
connected complement and empty interior. For further details see [5] and [6].

In this paper, we give the characterization of the structure of amenable
operators. At first, we use the reduction theory of von Neumann to give two
equivalent descriptions for Conjecture 1.1; and then, we give two
decompositions for amenable operators, which supporting the Conjecture 1.1.

##  2\. An equivalent formulation of the conjecture 1.1

In this section we use the reduction theory of von Neumann to give two
equivalent descriptions for Conjecture 1.1. We obtain that every amenable
operator is similar to a normal operator if and only if every non-scalar
amenable operator has a non-trivial hyperinvariant subspace if and only if
every amenable operator is similar to a reducible operator and has a non-
trivial invariant subspace.

In order to proof the main theorem, we need to introduce von Neumann’s
reduction theory [15] and some lemmas.

Let
$\mathfrak{H_{1}}\subseteq\mathfrak{H_{2}}\subseteq\cdots\subseteq\mathfrak{H_{\infty}}$
be a sequence of Hilbert spaces chosen once and for all, $\mathfrak{H_{n}}$
having the dimension $n$. Let $\mu$ be a finite positive regular measure
defined on the Borel sets of a separable metric space $\wedge$, and let
$\\{E_{n}\\}_{n=1}^{\infty}$ be a collection of disjoint Borel sets of
$\wedge$ with union $\wedge$. Then the symbol

$\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$

denotes the set of all functions $f$ defined on $\wedge$ such that

(1)$f(\lambda)\in\mathfrak{H}_{n}\subseteq\mathfrak{H}_{\infty}$ if
$\lambda\in E_{n}$;

(2)$f(\lambda)$ is a $\mu$-measurable function with values in
$\mathfrak{H}_{\infty}$;

(3) $\int_{\wedge}^{\oplus}|f(\lambda)|^{2}\mu(d\lambda)<\infty$.

We put

(4)$(f,g)=\int_{\wedge}^{\oplus}(f(\lambda),g(\lambda))\mu(d\lambda).$
The set of functions thus defined is called the direct integral Hilbert space
with measure $\mu$ and dimension sets $\\{E_{n}\\}$ and denoted by
$\mathfrak{H}=\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$.

An operator on $\mathfrak{H}$ is said to be decomposable if there exists a
strongly $\mu$-measurable operator-value function $A(\cdot)$ defined on
$\wedge$ such that $A(\lambda)$ is a bounded operator on the space
$\mathfrak{H}(\lambda)=\mathfrak{H}_{n}$ when $\lambda\in E_{n}$, and for all
$f\in\mathfrak{H}$, $(Af)(\lambda)=A(\lambda)f(\lambda)$. We write
$A=\int_{\wedge}^{\oplus}A(\lambda)\mu(d\lambda)$ for the equivalence class
corresponding to $A(\cdot)$. If $A(\lambda)$ is a scalar multiple of the
identity on $\mathfrak{H}(\lambda)$ for almost all $\lambda$, then $A$ is
called diagonal. The collection of all diagonal operator is called the
diagonal algebra of $\wedge$. In [15]I.3, Schwartz showed that an operator $A$
on Hilbert space
$\mathfrak{H}=\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$ is
decomposable if and only if $A$ belong to the commutant of the diagonal
algebra of $\wedge$. And $||A||=\mu-ess.sup_{\lambda\in\wedge}||A(\lambda)||$.

In [1], Azoff, Fong and Gilfeather used von Neumann’s reduction theory to
define the reduction theory for non-selfadjoint operator algebras: Fix a
partitioned measure space $\wedge$ and let $\mathfrak{D}$ be the corresponding
diagonal algebra. Given an algebra $\mathfrak{A}$ of decomposable operators.
Each operator $A\in\mathfrak{A}$ has a decomposition
$A=\int_{\wedge}^{\oplus}A(\lambda)\mu(d\lambda)$. Chosse a countable
generating set $\\{A_{n}\\}$ for $\mathfrak{A}$. let $\mathfrak{A}(\lambda)$
be the strongly closed algebra generated by the $\\{A_{n}(\lambda)\\}$.
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ is
called the decomposition of $\mathfrak{A}$ respect to $\mathfrak{D}$. A
decomposition
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ of
an algebra is said to be maximal if the corresponding diagonal algebra is
maximal among the abelian von Neumman subalgebras of $\mathfrak{A}^{\prime}$.
The following lemma is a basic result in [1] which will be used in this paper.

######  Lemma 2.1.

([1], Theorem 4.1) Let
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ be
a decomposition of a reductive algebra. Then almost all of
$\\{\mathfrak{A}(\lambda)\\}$ are reducible. In particular, if the
decomposition is maximal, then almost all of the algebras
$\\{\mathfrak{A}(\lambda)\\}$ are transitive.

In [7] Gifford studied the reduction property for operator algebras and
obtained the following result:

######  Lemma 2.2.

([7] Lemma 4.4, Lemma 4.12) If $\mathfrak{A}$ is a commutative amenable
operator algebra, then $\mathfrak{A}^{\prime},\mathfrak{A}^{\prime\prime}$ are
complete reducible and there exists $M\geq 1$ so that for any idempotent
$p\in\mathfrak{A}^{\prime\prime}$ $||p||\leq M$.

Assume $\mathfrak{A}$ is a operator algebra, let $P(\mathfrak{A})$ denote the
idempotents in $\mathfrak{A}$ and $\mathcal{P}(\mathfrak{A})$ denote the
strongly closed algebra generated by $P(\mathfrak{A})$. We get the following
lemma:

######  Lemma 2.3.

If $\mathfrak{A}$ is a commutative amenable operator algebra, then
$\mathcal{P}(\mathfrak{A^{\prime\prime}})$ is similar to an abelian von
Neumann algebra.

###### Proof.

By Lemma 2.2 and [2, Corollary 17.3], it follows that there exists
$X\in\mathfrak{B}(\mathfrak{H})$ such that $XpX^{-1}$ is selfadjoint for each
$p\in P(\mathfrak{A^{\prime\prime}})$. Hence
$\mathcal{P}(\mathfrak{A^{\prime\prime}})$ is similar to a abelian von Neumann
algebra. ∎

######  Lemma 2.4.

([5]) Let $\mathfrak{A}$ and $\mathfrak{B}$ be Banach algebras and suppose
that $\varphi:\mathfrak{A}\longrightarrow\mathfrak{B}$ is a continuous
homomorphism with $\varphi(\mathfrak{A})$ dense in $\mathfrak{B}$. If
$\mathfrak{A}$ is amenable, then $\mathfrak{B}$ is amenable.

######  Notation 2.5.

From Lemma 2.3,2.4 we always assume that
$\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$ is a abelian von Neumann
algebra, and $\mathfrak{A}_{T}^{\prime}$ is a reducible operator algebra in
this section.

Now we will proof the main result of this section:

######  Theorem 2.6.

The following are equivalent:

(1) Every amenable operator is similar to a normal operator;

(2) Every non-scalar amenable operator has a non-trivial hyperinvariant
subspace;

(3) Every amenable Banach algebra which is generated by an operator is similar
to a $C^{*}$-algebra.

###### Proof.

$(1)\Leftrightarrow(3)$ and $(1)\Rightarrow(2)$ is clear by [5]. Therefore, in
order to establish the theorem it suffices to show the implications
$(2)\Rightarrow(1)$.

Assume (2), by Lemma 2.3 choose a maximal decomposition for
$\mathfrak{A}_{T}^{\prime}\sim\int_{\wedge}^{\oplus}\mathfrak{A}_{T}^{\prime}(\lambda)\mu(d\lambda)$
respect to the diagonal algebra
$\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$.

Assume $T\sim\int_{\wedge}^{\oplus}T(\lambda)\mu(d\lambda)$ is the
decomposition for $T$. Let $\\{p_{n}\\}_{n=1}^{\infty}$ denote the all
rational polynomials. Then
$p_{n}(T)\sim\int_{\wedge}^{\oplus}p_{n}(T)(\lambda)\mu(d\lambda)$ is
decomposable for all $n$ and there exists a measurable $E\subseteq\wedge$ such
that $\mu(\wedge-E)=0$ and for any $\lambda\in E$ we have
$p_{n}(T)(\lambda)=p_{n}(T(\lambda))$ and
$||p_{n}(T)(\lambda)||\leq||p_{n}(T)||$ by [15, Lemma I.3.1, I.3.2]. Define a
mapping
$\varphi_{\lambda}:\mathfrak{A}_{T}\rightarrow\mathfrak{A}_{T(\lambda)}$ by
$\varphi_{\lambda}(p_{n}(T))=p_{n}(T(\lambda))$ for each rational polynomial
$p_{n}$ and $\lambda\in E$. Note that $||p_{n}(T(\lambda))||\leq||p_{n}(T)||$
for each rational polynomial $p_{n}$ and furthermore, $\\{p_{n}(T)\\}$ is
dense in $\mathfrak{A}_{T}$. Hence, $\varphi_{\lambda}$ is well-defined and
$\varphi_{\lambda}$ a continuous homomorphism with $\varphi(\mathfrak{A}_{T})$
dense in $\mathfrak{A}_{T(\lambda)}$. By Lemma 2.4, $T(\lambda)$ is amenable
for almost all $\lambda$.

Now for almost all $\lambda$, $T(\lambda)$ is amenable and
$\mathfrak{A}_{T}^{\prime}(\lambda)\subseteq\mathfrak{A}_{T(\lambda)}^{\prime}$
and $\mathfrak{A}_{T}^{\prime}(\lambda)$ is transitive by Lemma 2.1. Thus
almost all of $T(\lambda)$ are scalar operators, i.e. $T$ is a normal
operator. ∎

######  Corollary 2.7.

Every amenable operator is similar to a normal operator if and only if there
exists a non-trivial idempotent in the double-commutant of every non-scalar
amenable operator.

######  Remark 2.8.

In [5], Farenick, Forrest and Marcoux showed that if
$T\in\mathfrak{B}(\mathfrak{H})$ is amenable and similar to a normal operator
$N$, then the spectrum of $N$ has connected complement and empty interior.
According to [14, Theorem 1.23], $N$ is a reducible operator. Hence, there
exists an invertible operator $X\in\mathfrak{B}(\mathfrak{H})$ such that
$\mathfrak{A}_{XTX^{-1}}^{\prime\prime}$ is a reducible algebra. The following
theorem give the equivalent description for Conjecture 1.1 from the existence
of invariant subspace for amenable operators.

######  Theorem 2.9.

The following are equivalent:

(1) Every amenable operator is similar to a normal operator;

(2) For every amenable operator $T\in\mathfrak{B}(\mathfrak{H})$, there exists
an invertible operator $X\in\mathfrak{B}(\mathfrak{H})$ such that
$\mathfrak{A}_{XTX^{-1}}^{\prime\prime}$ is a reducible algebra and $T$ has a
non-trivial invariant subspace.

###### Proof.

$(1)\Rightarrow(2)$ is clear by Remark 2.8.

$(2)\Rightarrow(1)$ is trivial modifications adapt the proof of theorem 2.6. ∎

######  Remark 2.10.

According to theorem 2.6, 2.9, we obtain that the Conjecture 1.1 for operator
algebra which is generated by an operator is equivalent to the following
statements:

(1) Every amenable operator $T$ has a non-trivial invariant subspace and
renorm $\mathfrak{H}$ with an equivalent Hilbert space norm so that under this
norm Lat$\mathfrak{A}_{T}$ becomes orthogonally complemented;

(2) Every non-scalar amenable operator has a non-trivial hyperinvariant
subspace.

##  3\. Decomposition of Amenable operators

In this section, we get two decompositions for amenable operators and prove
that the two decompositions are the same which supporting Conjecture 1.1.

At first, we summarize some of the details of multiplicity theory for abelian
von Neumann algebras. For the most part, we will follow [3]. If $A$ is an
operator on a Hilbert space $\mathfrak{K}$ and $n$ is a cardinal number, Let
$\mathfrak{K}^{n}$ denote the orthogonal direct sum of $n$ copies of
$\mathfrak{K}$, and $A^{(n)}$ be the operator on $\mathfrak{K}^{n}$ which is
the direct sum of $n$ copies of $A$. Whenever $\mathfrak{A}$ is an operator
algebra on $\mathfrak{K}$, $\mathfrak{A}^{(n)}$ denotes the algebra
$\\{A^{(n)},A\in\mathfrak{A}\\}$. An abelian von Neumann algebra
$\mathfrak{B}$ is of uniform multiplicity $n$ if it is (unitary equivalent to)
$\mathfrak{A}^{(n)}$ for some maximal abelian von Neumann algebra
$\mathfrak{A}$. By [3], for any abelian von Neumann algebra $\mathfrak{A}$,
there exists a sequens of regular Borel measures $\\{\mu_{n}\\}$ on a sequens
of separable metric space $\\{X_{n}\\}$ such that $\mathfrak{A}$ is unitary
equivalent to
$\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}\oplus\mathfrak{B}_{\infty}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $1\leq n\leq\infty$. For further details see [3, II.3].

######  Proposition 3.1.

Suppose that $T$ is amenable operator and $\mathfrak{A}_{T}^{\prime}$ contains
a subalgebra which is similar to an abelian von Neumman algebra with no direct
summand of uniform multiplicity infinite, then $T$ is similar to a normal
operator.

###### Proof.

For the sake of simplicity, we assume $\mathfrak{A}_{T}^{\prime}$ contains a
subalgebra $\mathfrak{B}$ which is an abelian von Neumman subalgebra with no
direct summand of uniform multiplicity infinite. Trivial modifications adapt
the proof to the more general case.

By [3, II.3], there exists a sequens of regular Borel measures $\\{\mu_{n}\\}$
on a sequens of separable metric space $\\{X_{n}\\}$ such that $\mathfrak{B}$
is unitarily equivalent to $\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $n$. Hence, $T=\sum_{n=1}^{\infty}\oplus T_{n}$, where
$T_{n}\in\mathfrak{B}_{n}^{\prime}$. It suffices to show that $T_{n}$ is
similar to a normal operator for all $n$, then by [4, Corollary 26], it
follows that $T$ is similar to a normal operator.

Since $T\in\mathfrak{B}_{n}^{\prime}$, according to [14, Theorem 7.20], for
any $1\leq n<\infty$ there exists a unitary operator
$U_{n}\in\mathfrak{B}_{n}^{\prime}$ such that

$U_{n}T_{n}(U_{n})^{-1}=\left[\begin{array}[]{ccccc}N_{11}&N_{12}&\cdots&\cdots&N_{1n}\\\
0&N_{22}&\cdots&\cdots&N_{2n}\\\ 0&0&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&\vdots\\\ 0&0&\cdots&0&N_{nn}\\\
\end{array}\right]$

where $N_{ij}$ is a normal operator for all $1\leq i,j\leq n$. By [17,
Proposition 3.1], it follows that $T_{n}$ is similar to
$\oplus_{i=1}^{n}N_{ii}$, i.e. $T_{n}$ is similar to a normal operator for all
$n$. ∎

######  Corollary 3.2.

Assume $T$ is amenable operator, then there exists hyperinvariant subspaces
$M,N$ of $T$ such that $T$ has the form $T=T_{1}\dot{+}T_{2}$ respect to the
space decomposition $\mathfrak{H}=M\dot{+}N$, where $T_{1},T_{2}$ are amenable
operators, $T_{1}$ is similar to a normal operator and
$\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar to an abelian
von Neumman algebra with uniform multiplicity infinite.

The proof of the following lemma is straightforward and we omit it.

######  Lemma 3.3.

Suppose that $\mathfrak{A}$ is a completely reductive operator algebra and
$p\in P(\mathfrak{A}^{\prime})$. Then $p\mathfrak{A}$ is a completely
reductive operator algebra on $\textup{Ran}p$.

We are in need of the following propositions before we can address the main
theorem of this section.

######  Proposition 3.4.

Assume that $T$ is a amenable operator and there exists a space decomposition
$\mathfrak{H}=M\dot{+}N$ such that $T$ has the matrix form
$T=\left[\begin{array}[]{cc}T_{1}&\\\
&T_{2}\end{array}\right]\begin{matrix}\mbox{$M$}\\\ \mbox{$N$}\\\
\end{matrix}$. Then $T$ is similar to a normal operator if and only if $T_{1}$
and $T_{2}$ are similar to normal operators.

###### Proof.

Assume that $T$ has the matrix form

$T=\left[\begin{array}[]{cc}T_{1}&T_{12}\\\
&\widetilde{T_{2}}\end{array}\right]\begin{matrix}\mbox{$M$}\\\
\mbox{$M^{\bot}$}\\\ \end{matrix}$

respect to the space decomposition $\mathfrak{H}=M\oplus M^{\bot}$. By [17,
Lemma 2.8], there exists an invertible operator
$S=\left[\begin{array}[]{cc}I&S_{12}\\\
&I\end{array}\right]\begin{matrix}\mbox{$M$}\\\ \mbox{$M^{\bot}$}\\\
\end{matrix}$ such that $S^{-1}TS=\left[\begin{array}[]{cc}T_{1}&\\\
&\widetilde{T_{2}}\end{array}\right]\begin{matrix}\mbox{$M$}\\\
\mbox{$M^{\bot}$}\\\ \end{matrix}$. Assume that $S$ has the matrix form
$S=\begin{array}[]{cc}\begin{array}[]{cc}M&M^{\bot}\end{array}\\\
\left[\begin{array}[]{cc}I&\\\
&S_{1}\end{array}\right]\end{array}\begin{matrix}\\\ \mbox{$M$}\\\
\mbox{$N$}\\\ \end{matrix}$, we obtain that
$T_{2}=S_{1}\widetilde{T_{2}}S_{1}^{-1}$. By [9, propsition 6.5], we get that
$T$ is similar to a normal operator if and only if $T_{1}$ and $T_{2}$ are
similar to normal operators. ∎

######  Proposition 3.5.

Suppose that $T$ is an amenable operator, $M_{1}\in$
Lat$\mathfrak{A}_{T}^{\prime}$ and $M_{2}\in$
Lat$\mathfrak{A}_{T}^{\prime\prime}$. Then $M_{1}+M_{2}$ is closed.

Moreover, if $T|_{M_{1}}$ and $T|_{M_{2}}$ are similar to normal operators,
then $T|_{M_{1}+M_{2}}$ is similar to a normal operator.

###### Proof.

Let $N_{0}=M_{1}\cap M_{2}$, according to Lemma 3.3, there exists $N\in$
Lat$\mathfrak{A}_{T}^{\prime\prime}$ such that $M_{2}=N_{0}\dot{+}N$. Choose
$q\in P(\mathfrak{A}_{T}^{\prime})$, such that $\textup{Ran}q=N$. By the
assumption, $M_{1}\in$ Lat$\mathfrak{A}_{T}^{\prime}$. Hence $qM_{1}\subset
M_{1}\cap N=\\{0\\}$. Therefore $M_{1}\subset(I-q)\mathfrak{H}$. We see that
$M_{1}+M_{2}=M_{1}\dot{+}N$ is closed. This establishes the first statement of
the proposition.

Since $T|_{M_{2}}$ is similar to a normal operator, by proposition 3.4, we get
that $T|_{N}$ is similar to a normal operator. By the assumption $T|_{M_{1}}$
is similar to a normal operator, using proposition 3.4 again, we obtain that
$T|_{M_{1}+M_{2}}=T|_{M_{1}\dot{+}N}$ is similar to a normal operator. ∎

Now we will obtain the main theorem of this section.

######  Theorem 3.6.

Assume $T$ is an amenable operator, then there exists hyperinvariant subspaces
$M_{1},M_{2}$ of $T$ such that $T$ has the form $T=T_{1}\dot{+}T_{2}$ respect
to the space decomposition $\mathfrak{H}=M_{1}\dot{+}M_{2}$ and satisfies
that:

(1) $T_{1},T_{2}$ are amenable operators;

(2) If $M$ is a hyperinvariant subspace of $T$ and $T|_{M}$ is similar to a
normal operator, then $M\subseteq M_{1}$, i.e. $M_{1}$ is the largest
hyperinvariant subspace on which $T$ is similar to a normal operator;

(3) For any $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$,
$T_{2}|_{\textup{Ran}q}$ is not similar to a normal operator;

(4) $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite;

(5)
$\mathfrak{A}_{T}^{\prime}=\mathfrak{A}_{T_{1}}^{\prime}\dot{+}\mathfrak{A}_{T_{2}}^{\prime}$,
$\mathfrak{A}_{T}^{\prime\prime}=\mathfrak{A}_{T_{1}}^{\prime\prime}\dot{+}\mathfrak{A}_{T_{2}}^{\prime\prime}$;

(6) There exists no nonzero compact operator in
$\mathfrak{A}_{T_{2}}^{\prime}$.

###### Proof.

Case1. For any $p\in P(\mathfrak{A}_{T}^{\prime\prime})$, $T|_{\textup{Ran}p}$
is not similar to normal operator. According to the proof of Proposition 3.1,
we obtain that $\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite. Let $M_{1}=0$.

Case2. There exists $p\in P(\mathfrak{A}_{T}^{\prime\prime})$ such that
$T|_{\textup{Ran}p}$ is similar to normal operator. Then, by Zorn’s Lemma and
the same method in the proof of [4, Corollary 26], we can show that there
exists an element $p_{0}\in P(\mathfrak{A}_{T}^{\prime\prime})$ which is
maximal with respect to the property that $T|_{\textup{Ran}p_{0}}$ is similar
to a normal operator. Using Proposition 3.5, $\textup{Ran}p_{0}$ is the
largest hyperinvariant subspace of $T$ on which $T$ is similar to a normal
operator. Hence, $T$ has the form $T=T_{1}\dot{+}T_{2}$ with respect to the
space decomposition $\mathfrak{H}=\textup{Ran}p_{0}\dot{+}\textup{Ker}p_{0}$
where $T_{1}$ is similar to a normal operator, $T_{1},T_{2}$ are amenable
operators. Let $M_{1}=\textup{Ran}p_{0},M_{2}=\textup{Ker}p_{0}$.

Next we will prove that for any $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$,
$T_{2}|_{\textup{Ran}q}$ is not similar to normal operator. Then according to
Proposition 3.1 $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar
to an abelian von Neumman algebra with uniform multiplicity infinite.

Indeed, if there exists $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$ such
that $T_{2}|_{\textup{Ran}q}$ is similar to a normal operator and $q$ has the
form $q=\left[\begin{array}[]{cc}I&0\\\ 0&0\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}q$}\\\
\mbox{$\textup{Ker}q$}\\\ \end{matrix}$. Then for any
$A\in\mathfrak{A}_{T}^{\prime}$, $A$ has the form

$A=\left[\begin{array}[]{ccc}A_{11}&&\\\ &A_{22}&\\\ &&A_{33}\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}p_{0}$}\\\
\mbox{$\textup{Ran}q$}\\\ \mbox{$\textup{Ker}q$}\\\ \end{matrix}.$

Let

$R=\left[\begin{array}[]{ccc}I&&\\\ &I&\\\ &&0\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}p_{0}$}\\\
\mbox{$\textup{Ran}q$}\\\ \mbox{$\textup{Ker}q$}\\\ \end{matrix}.$

Then $R\in P(\mathfrak{A}_{T}^{\prime\prime})$. By the assumption
$T|_{\textup{Ran}R}$ is similar to a normal operator which contradicts to the
maximal property of $p_{0}$.

At last we will prove that there exists no nonzero compact operator in
$\mathfrak{A}_{T_{2}}^{\prime}$.

Indeed, if there exists a nonzero compact operator
$k_{0}\in\mathfrak{A}_{T_{2}}^{\prime}$, let $L_{1}$ denote the subspace
spanned by the ranges of all compact operators in
$\mathfrak{A}_{T_{2}}^{\prime}$, and $L_{2}$ the intersection of their kernel,
by [13, Lemma 3.1], both $L_{1},L_{2}$ lie in
Lat$\mathfrak{A}_{T_{2}}^{\prime}$ and
$L_{1}\dot{+}L_{2}=\textup{{Ker}}p_{0}$. Considering the restricting
$T_{2}|_{L_{1}}$, assume $T_{21}=T_{2}|_{L_{1}}$, then $T_{21}$ is an amenable
operator and $\mathfrak{A}_{T_{21}}^{\prime}$ contain a sufficient set of
compact operators. By Lemma 2.2 and [12, Theorem 9], $T_{21}$ is similar to a
normal operator which contradicts to the above discussion. ∎

Trivial modifications adapt the proof of Theorem 3.6, we obtain the following
theorem which decomposes amenable operators by the invariant subspaces of
them. The proof is similar to Theorem 3.6 and we omit it.

######  Theorem 3.7.

Assume $T$ is an amenable operator, then there exists invariant subspaces
$N_{1},N_{2}$ of $T$ such that $T$ has the form $T=A_{1}\dot{+}A_{2}$ respect
to the space decomposition $\mathfrak{H}=N_{1}\dot{+}N_{2}$ and satisfies
that:

(1) $A_{1},A_{2}$ are amenable operators;

(2) If $N$ is an invariant subspace of $T$ such that $N_{1}\subseteq N$ and
$T|_{N}$ is similar to a normal operator, then $N=N_{1}$, i.e. $N_{1}$ is the
maximal invariant subspace on which $T$ is similar to a normal operator;

(3) For any $q\in P(\mathfrak{A}_{T_{2}}^{\prime})$, $T_{2}|_{\textup{Ran}q}$
is not similar to a normal operator;

(4) If $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime})$ contains a subalgebra
which is similar to an abelian von Neumman algebra then the von Neumman
algebra has the uniform multiplicity infinite.

######  Remark 3.8.

If the answer to Conjecture 1.1 is positive, by Theorem 2.6, every amenable is
similar to a normal operator. Then, for the above theorem
$M_{1}=N_{1}=\mathfrak{H}$. That is to say, the two decompositions of theorem
3.6 and 3.7 are the same. The remainder of this section, we will prove that
the two decompositions are the same which supporting Conjecture 1.1.

######  Lemma 3.9.

[4] If $T\in\mathfrak{B}(\mathfrak{H})$ is an amenable operator and there
exist a one-to-one bounded linear map
$W:\mathfrak{H}\rightarrow\mathfrak{H}_{2}$, a bounded linear map
$V:\mathfrak{H}_{1}\rightarrow\mathfrak{H}$ with dense range and operators
$S_{1}\in\mathfrak{B}(\mathfrak{H}_{1})$,
$S_{2}\in\mathfrak{B}(\mathfrak{H}_{2})$ which are similar to normal operators
such that $TV=VS_{1}$ and $WT=S_{2}W$, then $T$ is similar to a normal
operator.

######  Corollary 3.10.

Assume $T=B_{1}B_{2}$ is an amenable operator, where $B_{1},B_{2}$ are
positive operators, then $T$ is similar to a normal operator.

###### Proof.

Assume $B_{1},B_{2}$ have the forms

$B_{2}=\left[\begin{array}[]{cc}0&\\\ &\widetilde{B_{2}}\\\
\end{array}\right],B_{1}=\left[\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{12}^{*}&B_{22}\\\ \end{array}\right],$

respect to the space decomposition
$\mathfrak{H}=\textup{Ker}B_{2}\oplus(\textup{Ker}B_{2})^{\bot}$ where
$\widetilde{B_{2}}$ is one-to-one and $B_{11},B_{22}$ are positive operators.
Thus $T$ has the form $T=\left[\begin{array}[]{cc}0&B_{12}\widetilde{B_{2}}\\\
0&B_{22}\widetilde{B_{2}}\\\ \end{array}\right]$ respect to the space
decomposition. Since $T$ is an amenable operator, by [17, Lemma 2.8], $T$ is
similar to $\left[\begin{array}[]{cc}0&0\\\ 0&B_{22}\widetilde{B_{2}}\\\
\end{array}\right]$. Thus without loss of generality, we may assume that
$B_{2}$ is one-to-one.

Assume that $B_{1},B_{2}$ has the form

$B_{1}=\left[\begin{array}[]{cc}\widetilde{B_{1}}&\\\ &0\\\
\end{array}\right],B_{2}=\left[\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{12}^{*}&B_{22}\\\ \end{array}\right],$

respect to the space decomposition
$\mathfrak{H}=(\textup{Ker}B_{1})^{\bot}\oplus\textup{Ker}B_{1}$ where
$\widetilde{B_{1}}$ is one-to-one and has dense range and $B_{11},B_{22}$ are
positive operators. Thus $T$ has the form
$T=\left[\begin{array}[]{cc}\widetilde{B_{1}}B_{11}&\widetilde{B_{1}}B_{12}\\\
0&0\\\ \end{array}\right]$ respect to the space decomposition. Since $T$ is an
amenable operator, by [17, Lemma 2.8], $T$ is similar to
$\left[\begin{array}[]{cc}\widetilde{B_{1}}B_{11}&0\\\ 0&0\\\
\end{array}\right]$ and there exists an operator $S$ such that
$\widetilde{B_{1}}B_{12}=\widetilde{B_{1}}B_{11}S$. Note that
$\widetilde{B_{1}},B_{2}$ are one-to-one, hence $B_{12}=B_{11}S$, and $B_{11}$
is one-to-one. Thus without loss of generality, we may assume that $B_{1}$ has
dense range and $B_{2}$ is one-to-one.

Note that
$B_{1}^{\frac{1}{2}}B_{2}B_{1}^{\frac{1}{2}},B_{2}^{\frac{1}{2}}B_{1}B_{2}^{\frac{1}{2}}$
are positive operators and
$TB_{1}^{\frac{1}{2}}=B_{1}^{\frac{1}{2}}B_{1}^{\frac{1}{2}}B_{2}B_{1}^{\frac{1}{2}}$
and
$B_{2}^{\frac{1}{2}}T=B_{2}^{\frac{1}{2}}B_{1}B_{2}^{\frac{1}{2}}B_{2}^{\frac{1}{2}}$,
by Lemma 3.9, $T$ is similar to a normal operator. ∎

######  Theorem 3.11.

The two decompositions for an amenable operator in Theorem 3.6, 3.7 are the
same.

###### Proof.

According to Theorem 3.6, 3.7, and Proposition 3.5, it is suffices to proof
that $N_{1}\in Lat\mathfrak{A}_{T}^{\prime}$.

In fact, if not. $T$ has the form $T=\left[\begin{array}[]{cc}T_{1}&\\\
&T_{2}\\\ \end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}$ and there exists $S=\left[\begin{array}[]{cc}0&0\\\ Y&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}\in\mathfrak{A}_{T}^{\prime}$ where $Y\neq 0$. Note that $S$ and
$T$ have the form

$S=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\ 0&0&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\
\mbox{$\overline{\textup{Ran}Y}$}\\\
\mbox{$N_{2}\ominus\overline{\textup{Ran}Y}$}\\\
\end{matrix},T=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&T_{22}\\\
&T_{23}&T_{24}\\\ \end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\
\mbox{$\overline{\textup{Ran}Y}$}\\\
\mbox{$N_{2}\ominus\overline{\textup{Ran}Y}$}\\\ \end{matrix},$

where $\tilde{Y}$ has dense range. Note that $TS=ST$, we get that $T_{23}=0$.
Since $T$ is amenable, by [17, Lemma 2.8] there exists an operator
$B:N_{2}\ominus\overline{\textup{Ran}Y}\rightarrow\overline{\textup{Ran}Y}$
such that

$\left[\begin{array}[]{ccc}I&0&0\\\ &I&B\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&T_{22}\\\
&&T_{24}\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-B\\\
&&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&0\\\
&&T_{24}\\\ \end{array}\right].$

Moreover,

$\left[\begin{array}[]{ccc}I&0&0\\\ &I&B\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\ 0&0&0\\\
\end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-B\\\ &&I\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\
0&0&0\\\ \end{array}\right].$

Hence, we can assume that $Y$ has dense range. Using $T$ is amenable again,
there exists $L=\left[\begin{array}[]{cc}0&X\\\ 0&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}\in\mathfrak{A}_{T}^{\prime}$, where $X\neq 0$, by [7, lemma
4.11]. Similar to the decomposition to $S$ and $T$, we get that $S$, $L$ and
$T$ have the form

$S=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y_{1}}&0&0\\\
\tilde{Y_{2}}&0&0\\\
\end{array}\right],L=\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right],T=\left[\begin{array}[]{ccc}T_{1}&0&0\\\
&T_{31}&T_{32}\\\ &T_{33}&T_{34}\\\ \end{array}\right],$

respect to the space decomposition
$\mathfrak{H}=N_{1}\oplus\textup{Ker}X\oplus(N_{2}\ominus\textup{Ker}X)$,
where $\tilde{X}$ is one-to-one, and $\tilde{Y_{1}},\tilde{Y_{2}}$ has dense
range. Note that $LT=TL$, we get that $T_{33}=0$. Using $T$ is amenable again,
there exists an operator $C:N_{2}\ominus\textup{Ker}X\rightarrow\textup{Ker}X$
such that

$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{31}&T_{32}\\\
&&T_{34}\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-C\\\
&&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{31}&0\\\
&&T_{34}\\\ \end{array}\right]$

$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-C\\\ &&I\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right]$

and

$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y_{1}}&0&0\\\
\tilde{Y_{2}}&0&0\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\
&I&-C\\\ &&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}0&0&0\\\
\tilde{Y_{1}}+C\tilde{Y_{2}}&0&0\\\ \tilde{Y_{2}}&0&0\\\ \end{array}\right].$

Moreover,
$\tilde{Y_{2}}T_{1}=T_{34}\tilde{Y_{2}},T_{1}\tilde{X}=\tilde{X}T_{34}$, and
$T_{1}$ is similar to a normal operator, by Lemma 3.9, $T_{34}$ is similar to
a normal operator, which contracts to Theorem 3.7. ∎

######  Corollary 3.12.

Assume $T$ is an amenable operator, then $M$ is a maximal invariant subspace
such that $T|_{M}$ is similar to a normal operator if and only if $M$ is the
largest invariant subspace such that $T|_{M}$ is similar to a normal operator.

######  Corollary 3.13.

Assume $T$ is an amenable operator and which is quasisimilar to a compact
operator, then $T$ is similar to a normal operator.

###### Proof.

Suppose, $TV=VK,WT=KW$ with $V,W$ injective operators having dense ranges and
$K$ is a compact operator. Then $TVKW=VKWT$. Let $C=VKW$,
$C\in\mathfrak{A}_{T}^{\prime}$, and $C$ is a compact operator. According to
Theorem 3.6, $C$ has the form $\left[\begin{array}[]{cc}C_{1}&\\\ &0\\\
\end{array}\right]$ respect to the space decomposition in the Theorem. If
$Cx=0$, $VWTx=Cx=0$, thus $Tx=0$. It follows that there is no part of $T_{2}$,
i.e. $T$ is similar to a normal operator. ∎

##  4\. (Essential) operator valued roots of abelian analytic functions

In this section, we will study the structure of an operator which is an
(essential) operator valued roots of abelian analytic functions and then we
get that if such an operator is also amenable, then it is similar to a normal
operator. In [8] Gilfeather introduce the concept of operator valued roots of
abelian analytic functions as follows: Let $\mathfrak{A}$ is an abelian von
Neumann algebra and $\psi(z)$, an $\mathfrak{A}$ valued analytic function on a
domain $\mathcal{D}$ in the complex plane. We may decompose $\mathfrak{A}$
into a direct integral of factors such that for $A\in\mathfrak{A}$, there
exists a unique $g\in L_{\infty}(\wedge,\mu)$ such that
$\mathfrak{A}=\int_{\wedge}^{\oplus}g(\lambda)I(\lambda)\mu(d\lambda)$. If
$T\in\mathfrak{A}^{\prime}$ and $\sigma(T)\subseteq\mathcal{D}$, let

$\psi(T)=(2\pi i)^{-1}\int_{\wedge}(T-zI)^{-1}\psi(z)dz.$

An operator $T$ is called a (essential)roots of the abelian analytic function
$\psi$, if $\psi(T)=0$(compact, respectively). The structure of roots of a
locally nonzero abelian analytic function has been given in [8], in this
section we main study the structure of essential roots of a locally nonzero
abelian analytic function.

######  Lemma 4.1.

Assume $T\in\mathfrak{B}(\mathfrak{H})$, $f$ is a locally nonzero analytic
function on the neighborhood of $\sigma(T)$ and assume $f(T)$ is a compact
operator, then $T$ is a polynomial compact operator.

###### Proof.

Let $\widehat{T}$ denote the image of $T$ in the Calkin algebra, then
$\widehat{f(T)}=0$. Since $f$ is a locally nonzero analytic function on
$\sigma(T)$, there exists a polynomial $p$ such that $\widehat{p(T)}=0$. i.e.
$T$ is a polynomial compact operator. ∎

######  Theorem 4.2.

Let $\psi$ be a locally nonzero abelian analytic function on $\mathcal{D}$
taking values in the von Neumann algebra $\mathfrak{A}$. If $T$ is an
essential roots of $\psi$ and is amenable, then $T$ is similar to a normal
operator.

###### Proof.

Since $\mathfrak{A}$ is an abelian von Neumann algebra, $\mathfrak{A}$ is
unitarily equivalent to
$\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}\oplus\mathfrak{B}_{\infty}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $1\leq n\leq\infty$. Note $T\in\mathfrak{A}^{\prime}$ is an amenable
operator, thus $T=T_{1}\oplus T_{2}$, where $T_{1}$ is similar to a normal
operator, and
$T_{1}\in(\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n})^{\prime},T_{2}\in\mathfrak{B}_{\infty}^{\prime}$.
Let $\sigma_{1}(\sigma_{2})$ denote the continuous (atom, respectively) parts
of the spectrum of $\mathfrak{B}_{\infty}$, then
$\mathfrak{B}_{\infty}=\mathfrak{C}_{\infty}\oplus\mathfrak{D}_{\infty}$,
where $\mathfrak{C}_{\infty}$ and $\mathfrak{D}_{\infty}$ are uniform
multiplicity $\infty$ von Neumman algebra and
$\sigma(\mathfrak{C}_{\infty})=\sigma_{1},\sigma(\mathfrak{D}_{\infty})=\sigma_{2}$
and $T_{2}=T_{3}\oplus T_{4}$, where
$T_{3}\in\sigma(\mathfrak{C}_{\infty})^{\prime},T_{4}\in\sigma(\mathfrak{D}_{\infty})^{\prime}$.

Assume $\psi$ is a locally nonzero abelian analytic function on $\mathcal{D}$
and $\sigma(T)\subseteq\mathcal{D}$, then
$\psi(T)=\psi(T_{1})\oplus\psi(T_{3})\oplus\psi(T_{4})$, note that
$\psi(T_{3})$ is a compact operator and
$\sigma(\mathfrak{C}_{\infty})=\sigma_{1}$, so $\psi(T_{3})=0$. Since
$\mathfrak{D}_{\infty}$ are uniform multiplicity $\infty$ and
$\sigma(\mathfrak{D}_{\infty})=\sigma_{2}$, by Lemma 4.1, it follows that
$T_{4}$ is direct sum of polynomial compact operators. According to [8,
Theorem 2.1], there exists a sequence of mutually orthogonal projections
$\\{P_{n},Q_{m}\\}$ in $\mathfrak{A}$ with $I=\sum P_{n}+\sum Q_{m}$ so that
$T|_{P_{n}}$ is finite type spectral operator and $T|_{Q_{m}}$ is polynomial
compact operator. By [17, Theorem 3.5, 4.5], we get that $T$ is similar to a
normal operator. ∎

## References

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Machine Learning Approaches to Automated Mechanism Design for Public Project
Problem Dr. Mingyu Guo, Dr. Wei Zhang Doctor of Philosophy Computer Science
University of Adelaide School of Computer Science Computer Science Education
Research Faculty Name

Guanhua Wang

A thesis submitted for the degree of
DOCTOR OF PHILOSOPHY
The University of Adelaide

August 27, 2024

###### Abstract

Abstract Mechanism design is a central research branch in microeconomics. An
effective mechanism can significantly improve performance and efficiency of
social decisions under desired objectives, such as to maximize social welfare
or to maximize revenue for agents.

However, mechanism design is challenging for many common models including the
public project problem model which we study in this thesis. A typical public
project problem is a group of agents crowdfunding a public project (e.g.,
building a bridge). The mechanism will decide the payment and allocation for
each agent (e.g., how much the agent pays, and whether the agent can use it)
according to their valuations. The mechanism can be applied to various
economic scenarios, including those related to cyber security. There are
different constraints and optimized objectives for different public project
scenarios (sub-problems), making it unrealistic to design a universal
mechanism that fits all scenarios, and designing mechanisms for different
settings manually is a taxing job. Therefore, we explore automated mechanism
design (AMD) ([sandholm2003automated]) of public project problems under
different constraints.

In this thesis, we focus on the public project problem, which includes many
sub-problems (excludable/non-excludable, divisible/indivisible, binary/non-
binary). We study the classical public project model and extend this model to
other related areas such as the zero-day exploit markets. For different sub-
problems of the public project problem, we adopt different novel machine
learning techniques to design optimal or near-optimal mechanisms via automated
mechanism design.

###### keywords:

###### Contents

  1. 1 List of Abbreviations
  2. 2 Publications

We evaluate our mechanisms by theoretical analysis or experimentally comparing
our mechanisms against existing mechanisms. The experiments and theoretical
results show that our mechanisms are better than state-of-the-art automated or
manual mechanisms.

##  Chapter 1 List of Abbreviations

List of Abbreviations

  * AMD

Automated mechanism design

  * GAN

Generative adversarial network

  * MLP

Multi-layer perceptions

  * AMA

Affine maximizer auctions ([Likhodedov2005:Approximating])

  * LP

Linear programming

  * IR

Individual rationality

  * SP

Strategy-proofness

  * SF

Straight-forwardness (Assumption LABEL:SF)

  * DP

Dynamic programming

  * PORF

Price-oriented rationing-free ([yokoo2002price])

  * DSIC

Dominant-strategy incentive compatible

  * CEC

Conservative equal cost mechanism

  * CDF

Cumulative distribution function

  * PDF

Probability density function

  * SCS

Serial cost sharing mechanism

  * NN

Neural network

  * TGA

Truthful genetic algorithm (In Chapter
LABEL:Public_Project_with_Minimum_Expected_Release_Delay)

  * ATGA

Approximately truthful genetic algorithm (In Chapter
LABEL:Public_Project_with_Minimum_Expected_Release_Delay)

  * VCG

Vickrey-Clarke-Groves mechanism

  * UB

Upper bound

I certify that this work contains no material which has been accepted for the
award of any other degree or diploma in my name, in any university or other
tertiary institution and, to the best of my knowledge and belief, contains no
material previously published or written by another person, except where due
reference has been made in the text. In addition, I certify that no part of
this work will, in the future, be used in a submission in my name, for any
other degree or diploma in any university or other tertiary institution
without the prior approval of the University of Adelaide and where applicable,
any partner institution responsible for the joint-award of this degree.

I acknowledge that copyright of published works contained within this thesis
resides with the copyright holder(s) of those works.

I also give permission for the digital version of my thesis to be made
available on the web, via the University’s digital research repository, the
Library Search and also through web search engines, unless permission has been
granted by the University to restrict access for a period of time.

Guanhua Wang

December 2021

###### Acknowledgements.

Firstly, I want to thank my supervisor Dr. Mingyu Guo. I am honorable working
with him in these three years. I very much appreciate his help. When I stock
in no research inspiration, he can always figure it out and find a suitable
way for me to solve the problems. Mingyu is a humble scholar. His research
foresight and research inspiration is lifelong treasure for me.
I would thank Dr. Wei Zhang, and professor Muhammad Ali Babar. They provide
some suggestions and comments that helped me to significantly improve the
content of this thesis and papers.
And from my Ph.D. role, I also thank my collaborators, lab meta, and my
friends for the help with the papers. I would like to mention Ba-Dung Le,
Runqi Guo, Yuko Sakurai, Wuli Zuo, Xie Yue, Zhigang Lu for many useful
discussions. My study area is not a hot area in this university. Thanks, you
give me many useful discussions. Good luck to you in the future. And I will
thank Wuli Zuo for checking and correcting my grammar mistakes.
I also want to thank my parents and my grandparents for financially supporting
me during my Ph.D. study. And because of Covid-19, it is a hard time to
connect you only on the internet. Hopefully, we can reunion soon after
Covid-19.
Finally, I will thank the university and its staff. They provide a good
environment for my research.

##  Chapter 2 Publications

This thesis is based on the following research papers that have been published
in peer-reviewed conferences or journals:

  * •

Guanhua Wang, Runqi Guo, Yuko Sakurai, Muhammad Ali Babar and Mingyu Guo.
Mechanism Design for Public Projects via Neural Networks. In Proceedings of
the 20th International Conference on Autonomous Agents and Multiagent Systems
(AAMAS 2021, Conference). (CORE Rank: A*)

  * •

Guanhua Wang, Mingyu Guo. Public Project with Minimum Expected Release Delay.
In the 18th Pacific Rim International Conference on Artificial Intelligence
(PRICAI, Conference) 2021. (CORE Rank: B)

  * •

Guanhua Wang, Wuli Zuo, Mingyu Guo. Redistribution in Public Project Problems
via Neural Networks. In the 20th IEEE/WIC/ACM International Conference on Web
Intelligence and Intelligent Agent Technology (WI-IAT, Conference), Melbourne,
Australia, 2021. (CORE Rank: B)

  * •

Mingyu Guo, Guanhua Wang, Hideaki Hata, M. Ali Babar. Revenue-Maximizing
Markets for Zero-Day Exploits. In Autonomous Agents and Multi-Agent Systems
(AAMAS, Journal) 35, 36, 2021. (CORE Rank: A)

Other research paper that has been published in peer-reviewed conferences but
not used in this thesis:

  * •

Ba-Dung Le; Guanhua Wang; Mehwish Nasim; Muhammad Ali Babar, "Gathering Cyber
Threat Intelligence from Twitter Using Novelty Classification," 2019
International Conference on Cyberworlds (CW, Conference), 2019, pp. 316-323,
(CORE Rank: B)