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Title: Weakly Supervised Clustering by Exploiting Unique Class Count

Paper content:
A weakly supervised learning based clustering framework is proposed in this paper.
As the core of this framework, we introduce a novel multiple instance learning task based on a bag level label called unique class count (ucc), which is the number of unique classes among all instances inside the bag.
In this task, no annotations on individual instances inside the bag are needed during training of the models.
We mathematically prove that with a perfect ucc classifier, perfect clustering of individual instances inside the bags is possible even when no annotations on individual instances are given during training.
We have constructed a neural network based ucc classifier and experimentally shown that the clustering performance of our framework with our weakly supervised ucc classifier is comparable to that of fully supervised learning models where labels for all instances are known.
Furthermore, we have tested the applicability of our framework to a real world task of semantic segmentation of breast cancer metastases in histological lymph node sections and shown that the performance of our weakly supervised framework is comparable to the performance of a fully supervised Unet model.
In machine learning, there are two main learning tasks on two ends of scale bar: unsupervised learning and supervised learning.
Generally, performance of supervised models is better than that of unsupervised models since the mapping between data and associated labels is provided explicitly in supervised learning.
This performance advantage of supervised learning requires a lot of labelled data, which is expensive.
Any other learning tasks reside in between these two tasks, so are their performances.
Weakly supervised learning is an example of such tasks.
There are three types of supervision in weakly supervised learning: incomplete, inexact and inaccurate supervision.
Multiple instance learning (MIL) is a special type of weakly supervised learning and a typical example of inexact supervision (Zhou, 2017) .
In MIL, data consists of bags of instances and their corresponding bag level labels.
Although the labels are somehow related to instances inside the bags, the instances are not explicitly labeled.
In traditional MIL, given the bags and corresponding bag level labels, task is to learn the mapping between bags and labels while the goal is to predict labels of unseen bags (Dietterich et al., 1997; Foulds & Frank, 2010) .
In this paper, we explore the feasibility of finding out labels of individual instances inside the bags only given the bag level labels, i.e. there is no individual instance level labels.
One important application of this task is semantic segmentation of breast cancer metastases in histological lymph node sections, which is a crucial step in staging of breast cancer (Brierley et al., 2016) .
In this task, each pathology image of a lymph node section is a bag and each pixel inside that image is an instance.
Then, given the bag level label that whether the image contains metastases or not, the task is to label each pixel as either metastases or normal.
This task can be achieved by asking experts to exhaustively annotate each metastases region in each image.
However, this exhaustive annotation process is tedious, time consuming and more importantly not a part of clinical workflow.
In many complex systems, such as in many types of cancers, measurements can only be obtained at coarse level (bag level), but information at fine level (individual instance level) is of paramount importance.
To achieve this, we propose a weakly supervised learning based clustering framework.
Given a dataset consisting of instances with unknown labels, our ultimate objective is to cluster the instances in this dataset.
To achieve this objective, we introduce a novel MIL task based on a new kind of bag level label called unique class count (ucc) , which is the number of unique classes or the number of clusters among all the instances inside the bag.
We organize the dataset into non-empty bags, where each bag is a subset of individual instances from this dataset.
Each bag is associated with a bag level ucc label.
Then, our MIL task is to learn mapping between the bags and their associated bag level ucc labels and then to predict the ucc labels of unseen bags.
We mathematically show that a ucc classifier trained on this task can be used to perform unsupervised clustering on individual instances in the dataset.
Intuitively, for a ucc classifier to count the number of unique classes in a bag, it has to first learn discriminant features for underlying classes.
Then, it can group the features obtained from the bag and count the number of groups, so the number of unique classes.
Our weakly supervised clustering framework is illustrated in Figure 1 .
It consists of a neural network based ucc classifier, which is called as Unique Class Count (U CC) model, and an unsupervised clustering branch.
The U CC model accepts any bag of instances as input and uses ucc labels for supervised training.
Then, the trained U CC model is used as a feature extractor and unsupervised clustering is performed on the extracted features of individual instances inside the bags in the clustering branch.
One application of our framework is the semantic segmentation of breast cancer metastases in lymph node sections (see Figure 4) .
The problem can be formulated as follows.
The input is a set of images.
Each image (bag) has a label of ucc1 (image is fully normal or fully metastases) or ucc2 (image is a mixture of normal and metastases).
Our aim is to segment the pixels (instances) in the image into normal and metastases.
A U CC model can be trained to predict ucc labels of individual images in a fully supervised manner; and the trained model can be used to extract features of pixels (intances) inside the images (bags).
Then, semantic segmentation masks can be obtained by unsupervised clustering of the pixels (each is represented by the extracted features) into two clusters (metastases or normal).
Note that ucc does not directly provide an exact label for each individual instance.
Therefore, our framework is a weakly supervised clustering framework.
Finally, we have constructed ucc classifiers and experimentally shown that clustering performance of our framework with our ucc classifiers is better than the performance of unsupervised models and comparable to performance of fully supervised learning models.
We have also tested the performance of our model on the real world task of semantic segmentation of breast cancer metastases in lymph node sections.
We have compared the performance of our model with the performance of popular medical image segmentation architecture of U net (Ronneberger et al., 2015) and shown that our weakly supervised model approximates the performance of fully supervised U net model 1 .
Hence, there are three main contributions of this paper:
1. We have defined unique class count as a bag level label in MIL setup and mathematically proved that a perfect ucc classifier, in principle, can be used to perfectly cluster the individual instances inside the bags.
2. We have constructed a neural network based ucc classifier by incorporating kernel density estimation (KDE) (Parzen, 1962) as a layer into our model architecture, which provided us with end-to-end training capability.
3. We have experimentally shown that clustering performance of our framework is better than the performance of unsupervised models and comparable to performance of fully supervised learning models.
The rest of the paper is organized such that related work is in Section 2, details of our weakly supervised clustering framework are in Section 3, results of the experiments on MNIST, CIFAR10 and CIFAR100 datasets are in Section 4, the results of the experiments in semantic segmentation of breast cancer metastases are in Section 5, and Section 6 concludes the paper.
In this paper, we proposed a weakly supervised learning based clustering framework and introduce a novel MIL task as the core of this framework.
We defined ucc as a bag level label in MIL setup and mathematically proved that a perfect ucc classifier can be used to perfectly cluster individual instances inside the bags.
We designed a neural network based ucc classifer and experimentally showed that clustering performance of our framework with our ucc classifiers are better than the performance of unsupervised models and comparable to performance of fully supervised learning models.
Finally, we showed that our weakly supervised unique class count model, U CC segment , can be used for semantic segmentation of breast cancer metastases in histological lymph node sections.
We compared the performance of our model U CC segment with the performance of a U net model and showed that our weakly supervised model approximates the performance of fully supervised U net model.
In the future, we want to check the performance of our U CC segment model with other medical image datasets and use it to discover new morphological patterns in cancer that had been overlooked in traditional pathology workflow.

assistant: text: A weakly supervised learning based clustering framework performs comparable to that of fully supervised learning models by exploiting unique class count.