Hi Matthew, I'd like to pursue that canopy thought a little further and mix it in with your sub sampling idea. Optimizing can come later, once we figure out how to do mean-shift in M/R at all. How about this?
1. Each mean-shift iteration consists of a canopy clustering of all the points, with T1 set to the desired sampling resolution (h?) and T2 set to 1. This will create one canopy centered on each point in the input set which contains all of its neighbors that are close enough to influence its next position in its trajectory (the window?). 2. We then calculate the centroid of each canopy (that's actually done already by the canopy cluster reducer). Is this centroid not also the weighted average you desire for the next location of the point at its center? 3. As the computation proceeds, the canopies will collapse together as their various centroids move inside the T2=1 radius. At the point when all points have converged, the remaining canopies will be the mean-shift clusters (modes?) of the dataset and their contents will be the migrated points in each cluster. 4. If each original point is duplicated as its own payload, then the iterations will produce clusters of migrated points whose payloads are the final contents of each cluster. Can you wrap your mind around this enough to validate my assumptions? Jeff > -----Original Message----- > From: Matthew Riley [mailto:[EMAIL PROTECTED] > Sent: Monday, March 10, 2008 5:58 PM > To: mahout-dev@lucene.apache.org > Subject: Re: Google Summer of Code[esp. More Clustering] > > Hi Jeff- > > I think your "basin of attraction" understanding is right on. I also like > your ideas for distributing the mean-shift iterations by following a > canopy-style method. My intuition was a little different, and I would like > to hear your ideas on it: > > Just to make sure we're on the same page.... > Say we have 1 million point in our original dataset, and we want to > cluster > by mean-shift. At each iteration of mean-shift we subsample (say) 10,000 > points from the original dataset and follow the gradient of those points > to > the region of highest density (and as we saw from the paper, rather than > calculate the gradient itself we can equivalently compute the weighted > average of our subsampled points and move the centroid to that point). > This > part seems fairly straightforward to distribute - we just send a different > subsampled set to each processor and each processor returns the final > centroid for that set. > > The problem I see is that 10,000 points (or whatever value we choose), may > be too much for a single processor if we have to compute the distance to > every single point when we compute the weighted mean. My thought here was > to > exploit the fact that we're using a kernel function (gaussian, uniform, > etc.) in the weighted mean calculation and that kernel will have a set > radius. Because the radius is static, it may be easy to (quickly) identify > the points that we must consider in the calculation (i.e. those within the > radius) by using a locality sensitive hashing scheme, tuned to that > particular radius. Of course, the degree of advantage we get from this > method will depend on the data itself, but intuitively I think we will > usually see a dramatic improvement. > > Honestly, I should do more background work developing this idea, and > possibly try a matlab implementation to test the feasibility. This sounds > more like a research paper than something we should dive into immediately, > but I wanted to share the idea and get some feedback if anyone has > thoughts... > > Matt > > > On Mon, Mar 10, 2008 at 11:29 AM, Jeff Eastman <[EMAIL PROTECTED]> > wrote: > > > Hi Matthew, > > > > I've been looking over the mean-shift papers for the last several days. > > While the details of the math are still sinking in, it looks like the > > basic algorithm might be summarized thusly: > > > > Points in an n-d feature space are migrated iteratively in the direction > > of maxima in their local density functions. Points within a "basin of > > attraction" all converge to the same maxima and thus belong to the same > > cluster. > > > > A physical analogy might be(?): > > > > Gas particles in 3-space, operating with gravitational attraction but > > without momentum, would tend to cluster similarly. > > > > The algorithm seems to require that each point be compared with every > > other point. This might be taken to require each mapper to see all of > > the points, thus frustrating scalability. OTOH, Canopy clustering avoids > > this by clustering the clusters produced by the subsets of points seen > > by each mapper. K-means has the requirement that each point needs to be > > compared with all of the cluster centers, not points. It has a similar > > iterative structure over clusters (a much smaller, constant number) that > > might be employed. > > > > There is a lot of locality in the local density function window, and > > this could perhaps be exploited. If points could be pre-clustered (as > > canopy is often used to prime the k-means iterations), parallelization > > might be feasible. > > > > Are these observations within a "basin of attraction" to your > > understanding of mean-shift <grin>? > > > > Jeff > > > > > > > > -----Original Message----- > > From: Matthew Riley [mailto:[EMAIL PROTECTED] > > Sent: Thursday, March 06, 2008 11:46 AM > > To: mahout-dev@lucene.apache.org > > Subject: Re: Google Summer of Code[esp. More Clustering] > > > > Hey Jeff- > > > > I'm certainly willing to put some energy into developing implementations > > of > > these algorithms, and it's good to hear that you may be interested in > > guiding us in the right direction. > > > > Here are the references I learned the algorithms from- some are more > > detailed than others: > > > > Mean-Shift clustering was introduced here and this paper is a thorough > > reference: > > Mean-Shift: A Robust Approach to Feature Space Analysis > > http://courses.csail.mit.edu/6.869/handouts/PAMIMeanshift.pdf > > > > And here's a PDF with just guts of the algorithm outlined: > > homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/TUZEL1/MeanShift.pdf > > > > It looks like there isn't a definitive reference for the k-means > > approximation with randomized k-d trees, but there are promising results > > introduced here: > > > > Object retrieval with large vocabularies and fast spatial matching: > > > http://www.robots.ox.ac.uk/~vgg/publications/papers/philbin07.pdf*<http:// > www.robots.ox.ac.uk/%7Evgg/publications/papers/philbin07.pdf*> > > * > > And a deeper explanation of the technique here: > > > > Randomized KD-Trees for Real-Time Keypoint Detection: > > ieeexplore.ieee.org/iel5/9901/31473/01467521.pdf?arnumber=1467521 > > > > Let me know what you think. > > > > Matt > > > > On Thu, Mar 6, 2008 at 11:45 AM, Jeff Eastman <[EMAIL PROTECTED]> > > wrote: > > > > > Hi Matthew, > > > > > > As with most open source projects, "interest" is mainly a function of > > > the willingness of somebody to contribute their energy. Clustering is > > > certainly within the scope of the project. I'd be interested in > > > exploring additional clustering algorithms with you and your > > colleague. > > > I'm a complete noob in this area and it is always enlightening to work > > > with students who have more current theoretical exposures. > > > > > > Do you have some links on these approaches that you find particularly > > > helpful? > > > > > > Jeff > > > > > > -----Original Message----- > > > From: Matthew Riley [mailto:[EMAIL PROTECTED] > > > Sent: Wednesday, March 05, 2008 11:11 PM > > > To: mahout-dev@lucene.apache.org; [EMAIL PROTECTED] > > > Subject: Re: Google Summer of Code > > > > > > Hey everyone- > > > > > > I've been watching the mailing list for a little while now, hoping to > > > contribute once I became more familiar, but I wanted to jump in here > > now > > > and > > > express my interest in the Summer of Code project. I'm currently a > > > graduate > > > student in electrical engineering at UT-Austin working in computer > > > vision, > > > which is closely tied to many of the problems Mahout is addressing > > > (especially in my area of content-based retrieval). > > > > > > What can I do to help out? > > > > > > I've discussed some potential Mahout projects with another student > > > recently- > > > mostly focused around approximate k-means algorithms (since that's a > > > problem > > > I've been working on lately). It sounds like you guys are already > > > implementing canopy clustering for k-means- Is there any interest in > > > developing another approximation algorithm based on randomized > > kd-trees > > > for > > > high dimensional data? What about mean-shift clustering? > > > > > > Again, I would be glad to help in any way I can. > > > > > > Matt > > > > > > On Thu, Mar 6, 2008 at 12:56 AM, Isabel Drost <[EMAIL PROTECTED]> > > > wrote: > > > > > > > On Saturday 01 March 2008, Grant Ingersoll wrote: > > > > > Also, any thoughts on what we might want someone to do? I think > > it > > > > > would be great to have someone implement one of the algorithms on > > > our > > > > > wiki. > > > > > > > > Just as a general note, the deadline for applications: > > > > > > > > March 12: Mentoring organization application deadline (12 noon > > > PDT/19:00 > > > > UTC). > > > > > > > > I suppose we should identify interesing tasks until that deadline. > > As > > > a > > > > general guideline for mentors and for project proposals: > > > > > > > > http://code.google.com/p/google-summer-of-code/wiki/AdviceforMentors > > > > > > > > Isabel > > > > > > > > -- > > > > Better late than never. -- Titus Livius (Livy) > > > > |\ _,,,---,,_ Web: <http://www.isabel-drost.de> > > > > /,`.-'`' -. ;-;;,_ > > > > |,4- ) )-,_..;\ ( `'-' > > > > '---''(_/--' `-'\_) (fL) IM: <xmpp://[EMAIL PROTECTED]> > > > > > > > > >
