Unsupervised Learning Of An Atlas From Unlabeled Point-sets
Unsupervised learning of an atlas from unlabeledpoint-setsHaili ChuiAnand Rangarajan and Jie ZhangMedical Imaging GroupDepartment of CISER2 Tech.Univ. of FloridaLos Altos, CAGainesville, FLhailic@r2tech.comanand,jiezhang @cise.ufl.edu AbstractOne of the key challenges in deformable shape modeling is the problem of estimating a meaningful average or mean shape from a set of unlabeled shapes. We presenta new joint clustering and matching algorithm that is capable of computing such amean shape from multiple shape samples which are represented by unlabeled pointsets. An iterative bootstrap process is used wherein multiple shape sample point-setsare non-rigidly deformed to the emerging mean shape, with subsequent estimation ofthe mean shape based on these non-rigid alignments. The process is entirely symmetric with no bias toward any of the original shape sample point-sets. We believe thatthis method can be especially useful for creating atlases of various shapes present inmedical images. We have applied the method to create a mean shape from nine handsegmented 2D corpus callosum data sets and from ten hippocampal 3D point-sets.1 IntroductionThe study of deformable shapes has recently been a very active area in computer visionand medical imaging. Most of the effort has been primarily focused on understandingdeformable shapes using a statistical approach. Analysis of deformable shapes has inturn helped create automated segmentation tools. A statistical understanding of shapes,their representations and deformations is therefore vitally important in many segmentation tasks. Statistical shape analysis using active shape models (Cootes et al., 1995) is1
a representative example in this category. After the shapes of a certain single structurebut from multiple subjects are extracted, much information can be gained by identifying, measuring and characterizing the shapes. Through such analysis, it is hoped thatthe subtle differences between different groups or populations, e.g. a normal group vs adiseased group, can be discovered and made available to aid clinical diagnosis. Recentwork in brain image analysis (Guimond et al., 2000; Davatzikos, 1997; Thompson et al.,1997) can be seen as examples in this category.1.1 Shape representationsDeformable shapes can have different representations. Curves and surfaces of the shapeboundary can obviously be used (Sebastian et al., 2000; Tagare, 1999). Despite being anintuitively natural choice for shape representation, curves and surfaces are somewhat difficult to directly use in statistical analysis. Using the intrinsic parameterization is difficultwhen the goal is atlas creation. Using one of many extrinsic parameterizations typicallyleads to a mixed representation defined on point locations and spline coefficients. Statistical shape analysis in this representation requires learning a density function defined onlocations and coefficients.Other than curves or surfaces, another natural choice is to use point location information. A point-set representation has the main advantage that statistical shape analysis canbe well formulated (Cootes et al., 1995) in that space. Not surprisingly, many recent statistical methodologies for shape analysis (Bookstein, 1989; Cootes et al., 1995; Duta et al.,1999) use the point-set representation. Our work also focuses on using points to studydeformable shapes.1.2 Statistical shape analysis with unknown correspondenceGiven a set of deformable shapes represented by point-sets, basic statistical measures suchas the mean and the covariance enable us to quantify the set of shapes at the very leastup to second-order statistics. The mean point-set is usually a placeholder for a normalrepresentative shape. The covariance information, usually of the form of a high dimensional covariance matrix, further provides us with information about how the shapes candeform and vary from the mean shape (in a second-order statistical sense). By examining the covariance matrix’s eigen-vectors and eigen-values, we can also observe thedominant variations or deformations present in the shape group (Cootes et al., 1995).If higher-order statistical information is desired, recent techniques such as IndependentComponent Analysis (ICA) can be pressed into service.2
The primary technical challenge in using point-set representations of shapes is thecorrespondence problem. Computing a meaningful mean point-set (and then the covariance matrix) from multiple shape point-sets is only possible when the correspondencesbetween all the shape point-sets are known. Automated correspondence estimation inshape point-sets is a non trivial task because of several reasons.Typically, correspondences can be estimated once the point-sets are properly alignedwith appropriate spatial transformations. An adequate transformation would obviouslybe a deformation, i.e., a non-rigid spatial mapping, since the objects at hand are deformable. Solving for non-rigid deformations between point-sets with unknown correspondence is a hard problem (Chui and Rangarajan, 2000). In fact, many current methods only attempt to solve for rigid transformations, e.g. affine transformations, for thealignment (Duta et al., 1999). The correspondences resulting from these methods are,therefore, only rough approximations since they ignore the non-rigid aspect of the transformation. However, when dealing with point-sets, there is another aspect of the atlasproblem, which we believe is more fundamental, and has largely been neglected (in thepoint-set literature).1.3 Joint clustering and matchingThe problem lies in the fact that the shape sample point-sets may not be consistent in thefirst place, i.e., points in each set may not be positioned at corresponding locations. Due tothe vagaries of feature extraction, points in one set may not have exactly counterparts inanother point-set. Without taking into account this feature extraction and sampling issue,the resulting correspondence achieved by any alignment method (rigid or non-rigid) isdoomed from the start.We face a dilemma here. On the one hand, we need to compare the point-sets, possibly through alignment, to each other in order establish correspondence. On the otherhand, the alignment won’t be entirely successful unless the sample point-sets have already satisfied the consistency criterion. The dilemma clearly shows that any process ofusing fixed sample point-sets to do the alignment is flawed. The problem is graphicallydepicted in Figure 1.Finally, we also encounter the bias problem in atlas creation. Since we have more thantwo sample point-sets to align, a question that arises is: how do we align all the pointsets in a symmetric manner so that there is no bias toward any particular point-set? Oneway to circumvent this problem is to define a mean point-set and then align every samplepoint-set to the mean. We see this as an opportunity for mutual improvement instead3
Original PointsSubsampled Points Set 2Subsampled Point Set 1Overlay of Subsampled Point SetsFigure 1: The consistency problem. From left to right: i) the original point-set withdensely distributed points; ii, iii) two subsampled point-sets; iv) the overlay of the subsampled point-sets. Note the difference between the two subsampled point-sets in theoverlay.of a fixed unidirectional process. As explained later, our approach consists of mutualrefinement of the mean shape and the non-rigid deformations from the point-sets to theemerging mean.After surveying the fundamental problems in atlas creation, we summarize our approach. A joint clustering and matching approach is used to overcome the consistencyproblem. The matching is performed on movable cluster centers and not on the originalpoints. The cluster centers are also used to estimate and re-estimate the atlas or meanshape through non-rigid matching and averaging. The estimated variables comprise (i)the cluster centers, (ii) the atlas (mean shape), (iii) the non-rigid transformations from (to)the cluster centers to (from) the atlas. We have designed a joint clustering and matchingalgorithm to carry out this iterative process. Embedded within a deterministic annealingscheme, the algorithm gradually refines all three sets of variables until convergence.2 ReviewAfter discussing the underlying motivations and rationale behind our method, we take astep back and briefly summarize some representative previous research work in this area.We also focus on the differences between these methods and ours.The work presented in (Sebastian et al., 2000) is a representative method using an intrinsic curve parameterization to analyze deformable shapes. It is very close to previousefforts in (Tagare, 1999) and (Davatzikos, 1997) where intrinsic curve properties, like thecurvature and the arc length, are used to align curves. In (Sebastian et al., 2000), the align-4
ment is done in a pairwise manner. To somewhat reduce the bias introduced by this nonsymmetrical procedure, the sample curves take turns in being the reference curve (atlas).The authors considered the symmetric alignment of multiple sample shapes to an emerging mean shape to be “intractable.” We are going to argue the opposite view in this paper.As with other methods using intrinsic curve or surface representations, further statisticalanalysis on these representations are much more difficult than the point representationbut the payoff may be greater due to the use of intrinisic higher-order representations.The active shape model proposed in (Cootes et al., 1995) utilized points to representdeformable shapes. Their work successfully initiated the research efforts in buildingpoint distribution models to understand deformable shapes (Cootes et al., 1995; Wangand Staib, 2000). To be able to build such a model, a set of training data, which are agroup of shape samples with known correspondences, are required. The training data arenormally acquired through a more or less manual landmarking process where a persongoes through all the samples and attempts, to the best of his/her knowledge, to markcorresponding points on each sample. It is a rather tedious process and the accuracy islimited. Since the derived point shape distribution is only going to be as good as thetraining data, it is highly desirable to automate the training data generation process andto make it more accurate. Different methods have been proposed to attack this problem.Rigid alignment (Duta et al., 1999) can be performed to get a rough estimate of the correspondences.Bookstein pioneered the usage of non-rigid spatial mapping functions - specificallythin-plate splines, to analyze deformable shapes (Bookstein, 1989). Since this method islandmark-based, it avoids the correspondence problem but suffers from the typical problems besetting landmark methods. Note that landmark-based methods largely overcomethe consistency problem since the placement of corresponding points is driven by thevisual perception of experts and not by an automated algorithm.The work in (Duta et al., 1999) also uses 2D points to learn shape statistics. The methodis quite similar to the active shape model method (Cootes et al., 1995) except that moreattention has been paid to the training data (or shape sample point-sets) generation process. The shape samples are first represented as curves and aligned with a rigid transformation. One curve is chosen as the reference and points are uniformly placed on thatcurve to form a reference point-set. To determine the location of corresponding pointson the other curves, the extrinsic curvature information is compared. Points are allowedto slide along the curves until their curvatures are in better agreement with each other.This shape learning method improves the consistency of the shape samples under the assumption that the curvature information can be reliably computed for the shapes at hand.5
Since the extrinsic curvature is a rigid invariant, only a rigid mapping can be used. Theprocess is not symmetric.3 Methodology3.1 Intuitive description of our approachThe correspondence problem, the shape sample consistency problem and the mean shapeestimation problem are all inter-related to each other. Rather than treat each of them asa separate problem, we propose to regard them as three interlocking steps in a moregeneral framework wherein we can simultaneously achieve meaningful answers for allthree problems. We also believe that directly solving for non-rigid transformations, ratherthan using indirect information criteria or relying on curvature information, has manyadvantages in terms of efficiency and accuracy. By directly modeling the deformation,the results will be intuitively easier to evaluate. Using non-rigid transformations, ratherthan rigid transformations is also one of the key aspects in improving the estimation ofcorrespondence.The three inter-locking steps in our atlas estimation framework are (i) clustering, (ii)non-rigid mapping estimation and (iii) atlas (mean shape) estimation. The clustering stepallows the cluster centers to be repositioned (to ”slide” along the original points) so thatthey have a better chance to be consistent. The overall Euclidean distance between thecluster centers and the original point-set is the criterion for shape representation. Thecluster center sets are associated with the emerging mean shape using non-rigid deformations. Since there is only one mean, this also provides an indirect connection betweenthe sample cluster center sets. A spline deformation energy is the criterion for shapematching. With consistent sample cluster center sets positioned at corresponding locations, and with multiple non-rigid transformations accounting for the shape differencesbetween the samples and the emerging mean, we apply the transformations and recompute the atlas (mean shape) by averaging over the warped cluster center sets. The averageEuclidean distance of all warped cluster center sets to the mean shape is the criterion foratlas estimation.A good way to understand this process is through the following: if the shape sampleof interest looks like a curved structure, the sample points would then all fall on this”imaginary curve”. After clustering, the cluster centers from the sample points will alsobe constrained to roughly lie on the imaginary curve because otherwise they won’t be agood shape representation of the original sample points. So the curve-like shape structure6
will always be maintained. However, this doesn’t prevent the cluster centers to have theextra freedom of being able to ”slide” along the ”imaginary curve” so that they can bebetter positioned to satisfy the consistency criteria.3.2 Learning an atlas as a density estimation problemOur approach to learning an atlas is based on density estimation. The basic idea is thatthe atlas is that point-set which best explains all the data point-sets. Since the atlas andthe data are not in the space space, the above statement is unpacked to mean that the atlasbest explains each data point-set once it has been registered and warped into the spaceof the data point-set. Consequently, the non-rigid registration parameters which take theatlas onto the data have to be included in the estimation. The overall estimation problemcan be situated within a Bayesian maximum a posteriori (MAP) framework.A Gaussian mixture model (McLachlan and Basford, 1988) is used to model each datapoint-set. Since the feature point-sets are usually highly structured, we can expect themto cluster well. Each point-set has the same number of cluster centers. Correspondenceis automatic since the same index is used for the cluster centers in all point-sets and theatlas. While the indices are common, the locations of the cluster centers will obviously bedifferent. .Our notation is as follows. The data point-sets are denoted by Each point-set consists of points ! " # . Since a Gaussian mixture model is used to model each point-set, we define a set of cluster centers, one for each pointset. The cluster center point-sets are denoted by &%' ( )* , . Each %- consistsof cluster locations /. 012 345 " 67 . Note that there are 6 points in each %8 .The number of clusters are chosen to be the same in each %9 and in the atlas. The atlaspoint-set is denoted by : and consists of points ; 0 3 ? @ " 67 . We begin byspecifying the density function of each point-set. AB DC % FE /G HKIDJQP MLON 0 LONE 0F AB C . 0 G (1)In (1), AB R C %- SET G is a mixture model containing component densities AU C ./ 0 G . The occupancy probability which is different for each data point-set is denoted by E . Later, wespecialize the component density to a Gaussian and set the occupancy probability to uniform in order to simplify (the gargantuan) atlas estimation procedure. Having specifiedthe density function of the data, we turn to the specification of the density function of the7
individual cluster centers &% 7 @ . A % C : G H: Q A PQC . 0 A(; 0 G/C 0 LONP0 LON C ; 0 A . 0 G/C G (2)The density function in (2) does not have a straightforward interpretation in terms ofcanonical parametric densities. Nonetheless, the basic idea in (2) is that each % is related to : via a pair of functions A G . The function models the mapping from the clustercenter set %9 to the atlas : with the function in the reverse mapping role. (The reason for employing two functions and will become clearer as we proceed.) Since thepoints in each %8 and in : are in correspondence, the density function in (2) defines a pair of landmark matching problems (one for and one for ). We now specify regularization priors on and . A G HIn (3), the operator: A C C ! C C G O A" /G H: # (3)A C C ! DC C G determines the kind of regularization imposed. For example,!!could correspond to a thin-plate spline, a Gaussian radial basis function, etc. Each choiceof ! is in turn related to a kernel and a metric of the deformation from and to : . Having specified the density functions of , % and the pair A , we may use Bayes’ theoremand obtain the posterior. A :- &% &% C G HLON AU C % G A % C :- G A G A" G A G (4)Some liberties have been taken in deriving (4). We assume that there is a “uniform”prior on : which is technically impossible since each point ; 0 is defined on all '"3)("* . Itis possible to work around this problem by using a Gaussian density with a very largevariance. We now specialize the density AU C % G to a Gaussian mixture model: AU C % SE , G HKwhere AU C . 0@ , G HI JQL N MLON 0 O32 A.- E G0/ 1 0 1P -E 0 AB C . 0 , 0 GAB N. 0 G 4 055 AU (5). 0 G 6 (6)where 7 0 3 ? 67 is the set of cluster covariance matrices. For the sake of simplicity and ease of implementation, we immediately specialize to the case where the occu-8
Npancy probabilities are uniform ABE 0 H P G and the covariance matrices 0 are isotropic,diagonal and identical [ A 0 H G ]. Clearly, these choices can be questioned. The oc cupancy probability contains valuable information regarding the number of members ina given cluster. And, the covariance matrix gives us valuable information regarding theprincipal direction (tangent vector) at each cluster center. Since we are already estimating the cluster centers, the deformation between the point-sets and the atlas and finallythe atlas, we have elected not to excessively burden the computation (in terms of speedand quality of solutions) by also estimating the occupancy probability and the covariancematrix of each cluster.The MAP estimation problem can be finally written as: % H" QQ %LONC . 0 J J J H %QI J LON MLON ! Q A(; 0 G/C"6P# A : &% C G30 LON A.- EC; ! G7/ 1 AB. 0 G/C " #C C!As it stands, minimizing (7) is awkward due to the - & C C %LONC 2. 0 C 6" #&C C ! C C %IDJ MLON'! &(7)PL N0 O , form ap-pearing in the first term of the objective. This is a well known problem in Gaussianmixture modeling (Redner and Walker, 1984). The expectation-maximization (EM) algorithm (McLachlan and Basford, 1988) has been quite popular since it avoids a directminimization of the above cost function. Since the mixture likelihood is non-convex, theEM algorithm is usually executed many times with varying random initial conditions.We instead adopt a deterministic annealing approach (Rose et al., 1990; Yuille et al., 1994;Hofmann and Buhmann, 1997) which (as we shall argue) is especially well suited to theatlas estimation task.The main difference between the traditional EM algorithm for mixtures and a deterministic annealing algorithm is in the treatment of the isotropic variance parameter . Indeterministic annealing, the variance parameter is externally imposed rather than beingestimated from within. A temperature parameter ( H - as in simulated annealing (orMCMC) is gradually lowered from high values to low values. When the temperature (is high, the cluster centers congregate around the center of mass of the point-sets. As thetemperature is lowered, a series of symmetry-breaking “phase transitions” (Rose et al.,1990) occur during which the cluster centers progressively move away from the center ofmass and toward their more local members.9
Consider the following Gaussian mixture likelihood objective function: AB. G HI Q Q'! LONP3 0 LON , - C 2. 0 C 6 (8)The objective function in (8) is a straightforward mixture objective without any deformation prior. Now consider A . G H IQQP 0 MLON 0 LON C . 0 C ' A . IQQP 0 '! MLON 0 LON 0 (9) and the temperature parameter ( . ItThe objective function in (9) has a new variableturns out that" ( G H(10)AB. GP and & 0 LON satisfies 0 0 H (Hathaway, 1986; Yuille et al., 1994) and when( is identified with - . The new variable 0 is a membership variable indicating thewhendegree to which each point feature belongs to cluster center . 0 . The main convenienceresulting from using (9) rather than (8) is that (9) does not have the & '! &form in it. , Also, the term & 0 0 '! 0 is an entropy barrier function with ( being the temperature.We now specify the overall cost function for atlas estimation. We convert the varianceparameter into a deterministic annealing parameter and rewrite the cost function usingthe new membership variables ? O 7 , .A :- &% %Q G H A(:- % LON (11) /G where A : % "" #C C !Q P0 LON DC CC; 0 " # G H A . 0 G/CC C ! C CIDJQQPL N MLON 0 OP" Q" (L N0 OI J PQQ 0 C 7. 0 CC A ; 0 G 7. 0 CL N MLON 0 O 0 '! 0 (12)Pwhere the membership matrix entries 0 and satisfy the constraint & 0 LON 0 H We briefly go over the individual terms in this energy function and explain some of thenew variables. The first term&IDJ MLON&PL N0 O 0 C ). 0 C measures the average distance be-10
tween the sample points and the cluster centers. Minimization of this term will essentially force all the cluster centers to be as close as possible to the original sample points,P %thus maintaining the original shape. The second and third terms & LON & 0 LON C ; 0 AB. 0 G C ,&% LONPL N C A(; 0 G 0 O& . 0 C , model the deformations (both the forward , which warps thecluster center sets % to : , and the reverse / , which does the opposite) to align the cluster centers with the mean. We think it is possible to force the deformation to be conN sistent by requiring to be A G 5 as in (Christensen, 1999) but have not done so here. " C C ! & C C measures the amount of distortion or bending introThe fourth term C C ! C C duced to the space by the non-rigid warps and & . By penalizing this bending measure, the algorithm will effectively place the mean shape points to be somewhat similarto all shape samples, since otherwise it will require a larger amount of bending to warpany of the shape samples onto the mean. Since all cluster center sets share a commonmean point set : , consistency between the cluster centers is also achieved. The fifth term( &I J MLON&PL N0 O 0 '! 0 arises from deterministic annealing (Chui and Rangarajan, 2000)as explained above. The parameter( , termed the temperature, is used to control thefuzziness of the clustering membership matrices : higher the temperature, greaterthe fuzziness. The '! form of the barrier function effectively leads to the formationof Gaussian clusters. The temperature ( can be intepreted as a common gaussian mixture variance parameter (Chui and Rangarajan, 2000; Yuille et al., 1994). However, it ismanually controlled as opposed to being automatically adjusted. The fuzziness of themembership matrices is gradually reduced by slowly annealing ( in a pre-defined linearscheme. In the experiments, we have empirically observed that annealing the regulariza#tion parameter leads to improved solutions. This allows us to focus on first estimatingthe rigid transformation parameters and then the non-rigid transformations.The whole setup is demonstrated in Figure 2. For simplicity, it is illustrated with onlytwo sample point-sets A( H - G . Extending the setup to incorporate multiple point-sets isobvious since there is no bias toward either point-set in Figure 2. The figure also depictsa possible outlier estimation extension which has not been implemented in this work.Having specified this objective function with three groups of unknown variables, weconduct a grouped coordinate descent procedure to achieve the energy minimization.Essentially, an alternating algorithm is used to cycle between the updates of each of thethree groups of variables.The update steps can be summarized as the following:(i) Update the cluster center sets: We first estimate the membership matrices 0 H&P 0 0 LON11 0 3T (13)
Feature Point Set X1Feature Point Set X2m1mOutlier ClusterClusteringMatchable Cluster CentersMatchable Cluster CentersClusteringOutlier ClusterCluster Center Set V 2Cluster Center Set V 1Inverse f12Forward f1Forward f2Inverse f2Average Point Set ZNFigure 2: The super clustering-matching algorithm. Each original point-set ( and G isN Nbeing clustered down to a set of cluster centers ( and %% G . Each cluster setNN P P N in % , . N in % G to account for the possible spurious points inhas an outlier cluster (.Neach point-set. The rest of the cluster centers ( /. 0@ 3 H ,- 67 and /. 0& 3 H ,-D T67 Gare being matched to the average point-set :- 12
where 0 H5 1 for 31 67 and ! " # (14) Then we compute the cluster centers,&. 0 HIDJ LON 0 IDJ MLON 0 &- " A ; 0 Gfor 3T (15)Note that the cluster centers are determined both by the original sample points aswell as the common mean : . As mentioned before, the cluster center sets are linkedto each other through the mean set : which enables them to achieve consistency. Theupdate equation for ? is exact whereas the update equation for &%8 @ is approximate.P We have neglected the term & 0believe that the “inverse” term &A closed-form solution for &% LON C ; 0 AB. 0 G/C in computing &% . This is because weP0 LON C A(; 0 G /. 0 C is a sufficient constraint for estimation.P cannot be obtained if the term & 0 LON C ; 0 AB. 0 G C is takeninto account.(ii) Update the atlas (mean shape) point-set:; 0 HQ % LON A . 0 G for 3 H(16) ,-D " 6 PThis update equation for : is approximate since we have neglected the term & 0 LON C A(; 0 G P./ 0 C in computing the mean : . This is because we believe that the forward term & 0 LON C ; 0 A ./ 0 G C is a sufficient constraint for estimation. A closed-form solution for : cannot bePobtained if the term & 0 LON C A ; 0 G ./ 0 C is taken into account. To get an intital estimate of : , we initialize the algorithm by setting to be identity transformations.(iii) Landmark matching: Update the non-rigid transformations: H J' AQP0 LONC; 0 A. 0 /G C" #C C ! C C G 3(17)with a similar update for the “inverse” transformation . Since this is equivalent to a landmark-based approach, we can solve for and in closed form. Our approachcan therefore be viewed as an automated landmarking process. If diffeomorphisms are required for , a single closed form solution is usually not possible. Instead, we wouldhave to introduce a “time” parameter and construct an ODE to take each % onto : as in(Camion and Younes, 2001; Joshi and Miller, 2000). The pseudo-code of the algorithm isas follows.13
The Super Clustering-Matching Algorithm:#Initialize parameters ( H ( and .Initialize all membership matrices (e.g., use uniform matrix). Initialize all transformation matrices (e.g., use identity transformation). Initialize the average point-set : (e.g., use the center of mass of all point-sets).Begin A: Deterministic Annealing.Begin B: Alternating Update.Step 1: Update cluster centers &%9 based on current ? , & & and :- Step 2: Update average point-set : based on cluster centers &%' andStep 3a: Update transformations & based on current %8 and : .Step 3b: Update cluster memberships based on current &% .End BDecrease ( ( until ( 0 is reached.End AInstead of having the flavor of essentially a gradient descent algorithm, which can beplagued by many local minima, o
mean shape from multiple shape samples which are represented by unlabeled point-sets. An iterative bootstrap process is used wherein multiple shape sample point-sets are non-rigidly deformed to the emerging mean shape, with subsequent estimation of the mean shape based on these non-rigid alignments. The process is entirely symmet-
Unsupervised learning For about 40 years, unsupervised learning was largely ignored by the machine learning community - Some widely used definitions of machine learning actually excluded it. - Many researchers thought that clustering was the only form of unsupervised learning. It is hard to say what the aim of unsupervised learning is.
Since research in feature selection for unsupervised learning is relatively recent, we hope that this paper will serve as a guide to future researchers. With this aim, we 1. Explore the wrapper framework for unsupervised learning, 2. Identify the issues involved in developing a feature selection algorithm for unsupervised learning within this .
Decision Trees and Random Forest Linear Regression. What are some limitations of supervised learning? Today: Unsupervised Learning In unsupervised learning, the training data is unlabeled . An Example: Clustering. Unsupervised Learning 1 3 .
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unsupervised machine learning, boasting applications in network intrusion detection, healthcare and many others. Several methods have been developed in recent years, however, a satisfactory solution is still missing to the best of our knowledge. We present Random Histogram Forest an effective approach for unsupervised anomaly detection.
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