Weakly supervised classification of objects in images using soft ...

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4 Riwal Lefort, Ronan Fablet, Jean-Marc BoucherConsequently, given a constructed decision tree, a test sample will be passedtrough the tree and be assigned the class priors of the terminal it will reache.Let us denote by p mi the class priors at node m of the tree. The key aspectof the weakly supervised learning of the soft decision tree is the computation ofclass prior p mi at any node m. In the supervised case it consists in evaluating theproportion of each class at node m. In a weakly supervised learning context, realclasses are unknown and class proportions can not be easily assessed. We proposeto compute p mi as a weighted sum over priors {π ni } for all samples attached tonode m. For descriptor d, denoting x d n the instance value and considering thechildren node m 1 that groups together data such as {x d n} < S d , the followingfusion rule is then proposed: ∑pm1i ∝(π ni ) α (3){n}|{x d n } S d , the∑(π ni ) α (4){n}|{x d n }>S dThe considered power α weighs low-uncertainty samples, i.e. samples such thatclass priors closer to 1 should contribute more to the overall cluster mean p mi . Aninfinite exponent values resorts to assigning the class with the greatest prior overall samples in the cluster. In contrast, an exponent value close to zero withdrawsfrom the weighted sum low class prior. In practice, we typically set α to 0.8.This setting comes to give more importance to priors close to one. If α < 1, highclass priors are given a similar greater weight compared to low class priors. Ifα > 1, the closer to one the prior the greater the weight.Considering a random forest, the output from each tree t for a given testdata x is a prior vector p t = {p ti }. p ti is the prior for class i at the terminalnode reached for tree t. The overall probability that x is assigned to class i, i.e.posterior likelihood p(y = i|x), is then given by the mean:p(y = i|x) = 1 T∑p ti (5)Tt=1where y n = i denotes that sample x n is assigned to class i. A hard classificationresorts to selecting the most likely class according to posteriors (5).3 Iterative classificationIn this section, an iterative procedure that is applied to the training dataset issuggested. A naive version is first presented, and finally, a more robust versionthat avoids over training is proposed.3.1 Naive iterative procedureThe basic idea of the iterative scheme is that the class priors of the trainingsamples can be refined iteratively from the overall knowledge acquired by the
Weakly supervised classification of objects in images 5trained classifier such that these class priors finally converge to the real class ofthe training samples. The classifier at a given iteration can then be viewed as afilter that should reduce the noise or uncertainty on the class priors of trainingsamples. Note that this iterative method is only applied to the training dataset.Such an iterative procedure has previously been investigated in different contexts,especially with probabilistic generative classifier [15]. Theoretical resultsregarding convergence properties can hardly be derived [28] [29], though goodexperimental performances have been reported [30]. The major drawbacks of thisapproach are possible over-training effects and the propagation of early classificationerrors [5]. Bayesian models may contribute to solve for these over-trainingissues.Given an initial training data set T 1 = {x n, π 1 n} and M iterations,1. For m from 1 to M– Learn a classifier C m from T m.– Apply the classifier C m to T m.– Update T m+1 = {x n, πn m+1 } with πnm+1 ∝ πnp(x 1 n|y n = i, C m).2. Learn the final classifier using T M+1.Table 1. Naive iterative procedure for weakly supervised learning (IP1).The implementation of this naive iterative procedure proceeds as follows forweakly supervised learning. At iteration m given the weakly supervised dataset{x n , πn m }, a random forest C m can be learnt. The updated random forest couldbe used to process any training sample {x n , πn m } to provide an updates classprior π m+1 . This update of class prior π m+1 should exploit both the output ofthe random forest and the initial prior π 1 . Here, the updated priors are given by:π m+1n∝ π 1 np(x n |y n = i, C m ) where y n = i denotes the classe variable for samplen.This algorithm is sketched in Tab. 1. In the subsequent, this procedure willbe referred to as IP1 (Iterative Procedure 1).3.2 Randomization-based iterative procedure without over trainingA major issue with the above naive iterative procedure is that the random forestis repeatedly applied to the training data such that over-training effects may beexpected. Such over-training effects should be avoided.To this end, we propose a second iterative procedure. The key idea is toexploit a randomization-based procedure to distinguish at each iteration separatetraining and test subsets. More precisely, we proceed as follows. At iteration m,the training dataset T m = {x n , πn m } is randomly split into a training datasetT r m and a test dataset T t m according to a given proportion β. T r m is exploitedto build a weakly supervised random forest C m . Samples in T t m are passedthrough random forest C m and updated class priors are issued from the same ruleas previously: πnm+1 ∝ πnp(x 1 n |y n = i, C m ). β gives the proportion of trainingexamples in the training set T r m while the remainder (1 − β) training examples
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