Shooting somewhat in the dark here, but isn't this susceptible to the mean-field techniques that are used for variational integration of Bayesian probabilistic techniques (except that you are starting with a field representation rather than introducing it)? I can't say what the details are, but this does seem plausible.
Also, wouldn't a sampling of the points give a good approximation of the field? If so, then building an n-centroid approximation of the field should be possible for a sampled subset. Then given that approximation, you can perform the mean shift very much in parallel at which point you can repeat the approximation and be ready to roll for the next shift. Moreover, since the fields are additive, you can compute the field on all of the sub-sample due to a sub-sub-sample at each of the map nodes and then combine them in a reduce step. That would allow you to simplify wha tis likely the most expensive step of the approximation. The k-means pre-step approach is also likely to be be helpful, at least in relatively low dimensional problems. In high dimensional spaces, however, everything outside a cluster is approximately equidistant so using the k-means approximation for the field simply reduces to a very expensive form of k-means clustering, or a variant on mean-shift with a useless k-means add-on. On 3/10/08 5:57 PM, "Matthew Riley" <[EMAIL PROTECTED]> wrote: > 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]> >>>> >>> >>
