I read a a paper that mention max min clustering algorithm, but i don't really quite understand what this algorithm does. Googling "max min clustering algorithm" doesn't yield any helpful result. does anybody know what this algorithm mean? this is an excerpt of the paper:
Max-min clustering proceeds by choosing an observation at random as the first centroid c1, and by setting the set C of centroids to {c1}. During the ith iteration, ci is chosen such that it maximizes the minimum Euclidean distance between ci and observations in C. Max-min clustering is preferable to a density-based clustering algorithm (e.g. k-means) which would tend to select many examples from the dense group of non-seizure data points.
I don't quite understand the bolded part.
link to paper is here
We choose each new centroid to be as far as possible from the existing centroids. Here's some Python code.
def maxminclustering(observations, k):
observations = set(observations)
if k < 1 or not observations: return set()
centroids = set([observations.pop()])
for i in range(min(k - 1, len(observations))):
newcentroid = max(observations,
key=lambda observation:
min(distance(observation, centroid)
for centroid in centroids))
observations.remove(newcentroid)
centroids.add(newcentroid)
return centroids
This sounds a lot like the farthest-points heuristic for seeding k-means, but then not performing any k-means iterations at all.
This is a surprisingly simple, but quite effective strategy. Basically it will find a number of data points that are well spread out, which can make k-means converge fast. Usually, one would discard the first (random) data point.
It only works well for low values of k though (it avoids placing centroids in the center of the data set!), and it is not very favorable to multiple runs - it tends to choose the same initial centroids again.
K-means++ can be seen as a more randomized version of this. Instead of always choosing the farthes object, it chooses far objects with increased likelihood, but may at random also choose a near neighbor. This way, you get more diverse results when running it multiple times.
You can try it out in ELKI, it is named FarthestPointsInitialMeans. If you choose the algorithm SingleAssignmentKMeans, then it will not perform k-means iterations, but only do the initial assignment. That will probably give you this "MaxMin clustering" algorithm.
I have a huge journal with actions done by users (like, for example, moderating contents).
I would like to find the 'mass' actions, meaning the actions that are too dense (the user probably made those actions without thinking it too much :) ).
That would translate to clustering the actions by date (in a linear space), and to marking the clusters that are too dense.
I am no expert in clustering algorithms and methods, but I think the k-means clustering would not do the trick, since I don't know the number of clusters.
Also, ideally, I would also like to 'fine tune' the algorithm.
What would you advice?
P.S. Here are some resources that I found (in Ruby):
hierclust - a simple hierarchical clustering library for spatial data
AI4R - library that implements some clustering algorithms
K-means would probably do a good job as long as you're interested in an a priori known number of clusters. Since you don't you might consider reading about the LBG algorithm, which is based on k-means and is used in data compression for vector quantisation. It's basically iterative k-means which splits centroids after they converge and keeps splitting until you achieve an acceptable number of clusters.
On the other hand, since your data is one-dimensional, you could do something completely different.
Assume that you've got actions which took place at 5 points in time: (8, 11, 15, 16, 17). Let's plot a Gaussian for each of these actions with μ equal to the time and σ = 3.
Now let's see how a sum of values of these Gaussians looks like.
It shows a density of actions with a peak around 16.
Based on this observation I propose a following simple algorithm.
Create a vector of zeroes for the time range of interest.
For each action calculate the Gaussian and add it to the vector.
Scan the vector looking for values which are greater than the maximum value in the vector multiplied by α.
Note that for each action only a small section of the vector needs updates because values of a Gaussian converge to zero very quickly.
You can tune the algorithm by adjusting values of
α ∈ [0,1], which indicates how significant a peak of activity has to be to be noted,
σ, which affects the distance of actions which are considered close to each other, and
time periods per vector's element (minutes, seconds, etc.).
Notice that the algorithm is linear with regard to the number of actions. Moreover, it shouldn't be difficult to parallelise: split your data across multiple processes summing Gaussians and then sum generated vectors.
Have a look at density based clustering. E.g. DBSCAN and OPTICS.
This sounds like exactly what you want.
I have asked a question a few days back on how to find the nearest neighbors for a given vector. My vector is now 21 dimensions and before I proceed further, because I am not from the domain of Machine Learning nor Math, I am beginning to ask myself some fundamental questions:
Is Euclidean distance a good metric for finding the nearest neighbors in the first place? If not, what are my options?
In addition, how does one go about deciding the right threshold for determining the k-neighbors? Is there some analysis that can be done to figure this value out?
Previously, I was suggested to use kd-Trees but the Wikipedia page clearly says that for high-dimensions, kd-Tree is almost equivalent to a brute-force search. In that case, what is the best way to find nearest-neighbors in a million point dataset efficiently?
Can someone please clarify the some (or all) of the above questions?
I currently study such problems -- classification, nearest neighbor searching -- for music information retrieval.
You may be interested in Approximate Nearest Neighbor (ANN) algorithms. The idea is that you allow the algorithm to return sufficiently near neighbors (perhaps not the nearest neighbor); in doing so, you reduce complexity. You mentioned the kd-tree; that is one example. But as you said, kd-tree works poorly in high dimensions. In fact, all current indexing techniques (based on space partitioning) degrade to linear search for sufficiently high dimensions [1][2][3].
Among ANN algorithms proposed recently, perhaps the most popular is Locality-Sensitive Hashing (LSH), which maps a set of points in a high-dimensional space into a set of bins, i.e., a hash table [1][3]. But unlike traditional hashes, a locality-sensitive hash places nearby points into the same bin.
LSH has some huge advantages. First, it is simple. You just compute the hash for all points in your database, then make a hash table from them. To query, just compute the hash of the query point, then retrieve all points in the same bin from the hash table.
Second, there is a rigorous theory that supports its performance. It can be shown that the query time is sublinear in the size of the database, i.e., faster than linear search. How much faster depends upon how much approximation we can tolerate.
Finally, LSH is compatible with any Lp norm for 0 < p <= 2. Therefore, to answer your first question, you can use LSH with the Euclidean distance metric, or you can use it with the Manhattan (L1) distance metric. There are also variants for Hamming distance and cosine similarity.
A decent overview was written by Malcolm Slaney and Michael Casey for IEEE Signal Processing Magazine in 2008 [4].
LSH has been applied seemingly everywhere. You may want to give it a try.
[1] Datar, Indyk, Immorlica, Mirrokni, "Locality-Sensitive Hashing Scheme Based on p-Stable Distributions," 2004.
[2] Weber, Schek, Blott, "A quantitative analysis and performance study for similarity-search methods in high-dimensional spaces," 1998.
[3] Gionis, Indyk, Motwani, "Similarity search in high dimensions via hashing," 1999.
[4] Slaney, Casey, "Locality-sensitive hashing for finding nearest neighbors", 2008.
I. The Distance Metric
First, the number of features (columns) in a data set is not a factor in selecting a distance metric for use in kNN. There are quite a few published studies directed to precisely this question, and the usual bases for comparison are:
the underlying statistical
distribution of your data;
the relationship among the features
that comprise your data (are they
independent--i.e., what does the
covariance matrix look like); and
the coordinate space from which your
data was obtained.
If you have no prior knowledge of the distribution(s) from which your data was sampled, at least one (well documented and thorough) study concludes that Euclidean distance is the best choice.
YEuclidean metric used in mega-scale Web Recommendation Engines as well as in current academic research. Distances calculated by Euclidean have intuitive meaning and the computation scales--i.e., Euclidean distance is calculated the same way, whether the two points are in two dimension or in twenty-two dimension space.
It has only failed for me a few times, each of those cases Euclidean distance failed because the underlying (cartesian) coordinate system was a poor choice. And you'll usually recognize this because for instance path lengths (distances) are no longer additive--e.g., when the metric space is a chessboard, Manhattan distance is better than Euclidean, likewise when the metric space is Earth and your distances are trans-continental flights, a distance metric suitable for a polar coordinate system is a good idea (e.g., London to Vienna is is 2.5 hours, Vienna to St. Petersburg is another 3 hrs, more or less in the same direction, yet London to St. Petersburg isn't 5.5 hours, instead, is a little over 3 hrs.)
But apart from those cases in which your data belongs in a non-cartesian coordinate system, the choice of distance metric is usually not material. (See this blog post from a CS student, comparing several distance metrics by examining their effect on kNN classifier--chi square give the best results, but the differences are not large; A more comprehensive study is in the academic paper, Comparative Study of Distance Functions for Nearest Neighbors--Mahalanobis (essentially Euclidean normalized by to account for dimension covariance) was the best in this study.
One important proviso: for distance metric calculations to be meaningful, you must re-scale your data--rarely is it possible to build a kNN model to generate accurate predictions without doing this. For instance, if you are building a kNN model to predict athletic performance, and your expectation variables are height (cm), weight (kg), bodyfat (%), and resting pulse (beats per minute), then a typical data point might look something like this: [ 180.4, 66.1, 11.3, 71 ]. Clearly the distance calculation will be dominated by height, while the contribution by bodyfat % will be almost negligible. Put another way, if instead, the data were reported differently, so that bodyweight was in grams rather than kilograms, then the original value of 86.1, would be 86,100, which would have a large effect on your results, which is exactly what you don't want. Probably the most common scaling technique is subtracting the mean and dividing by the standard deviation (mean and sd refer calculated separately for each column, or feature in that data set; X refers to an individual entry/cell within a data row):
X_new = (X_old - mu) / sigma
II. The Data Structure
If you are concerned about performance of the kd-tree structure, A Voronoi Tessellation is a conceptually simple container but that will drastically improve performance and scales better than kd-Trees.
This is not the most common way to persist kNN training data, though the application of VT for this purpose, as well as the consequent performance advantages, are well-documented (see e.g. this Microsoft Research report). The practical significance of this is that, provided you are using a 'mainstream' language (e.g., in the TIOBE Index) then you ought to find a library to perform VT. I know in Python and R, there are multiple options for each language (e.g., the voronoi package for R available on CRAN)
Using a VT for kNN works like this::
From your data, randomly select w points--these are your Voronoi centers. A Voronoi cell encapsulates all neighboring points that are nearest to each center. Imagine if you assign a different color to each of Voronoi centers, so that each point assigned to a given center is painted that color. As long as you have a sufficient density, doing this will nicely show the boundaries of each Voronoi center (as the boundary that separates two colors.
How to select the Voronoi Centers? I use two orthogonal guidelines. After random selecting the w points, calculate the VT for your training data. Next check the number of data points assigned to each Voronoi center--these values should be about the same (given uniform point density across your data space). In two dimensions, this would cause a VT with tiles of the same size.That's the first rule, here's the second. Select w by iteration--run your kNN algorithm with w as a variable parameter, and measure performance (time required to return a prediction by querying the VT).
So imagine you have one million data points..... If the points were persisted in an ordinary 2D data structure, or in a kd-tree, you would perform on average a couple million distance calculations for each new data points whose response variable you wish to predict. Of course, those calculations are performed on a single data set. With a V/T, the nearest-neighbor search is performed in two steps one after the other, against two different populations of data--first against the Voronoi centers, then once the nearest center is found, the points inside the cell corresponding to that center are searched to find the actual nearest neighbor (by successive distance calculations) Combined, these two look-ups are much faster than a single brute-force look-up. That's easy to see: for 1M data points, suppose you select 250 Voronoi centers to tesselate your data space. On average, each Voronoi cell will have 4,000 data points. So instead of performing on average 500,000 distance calculations (brute force), you perform far lesss, on average just 125 + 2,000.
III. Calculating the Result (the predicted response variable)
There are two steps to calculating the predicted value from a set of kNN training data. The first is identifying n, or the number of nearest neighbors to use for this calculation. The second is how to weight their contribution to the predicted value.
W/r/t the first component, you can determine the best value of n by solving an optimization problem (very similar to least squares optimization). That's the theory; in practice, most people just use n=3. In any event, it's simple to run your kNN algorithm over a set of test instances (to calculate predicted values) for n=1, n=2, n=3, etc. and plot the error as a function of n. If you just want a plausible value for n to get started, again, just use n = 3.
The second component is how to weight the contribution of each of the neighbors (assuming n > 1).
The simplest weighting technique is just multiplying each neighbor by a weighting coefficient, which is just the 1/(dist * K), or the inverse of the distance from that neighbor to the test instance often multiplied by some empirically derived constant, K. I am not a fan of this technique because it often over-weights the closest neighbors (and concomitantly under-weights the more distant ones); the significance of this is that a given prediction can be almost entirely dependent on a single neighbor, which in turn increases the algorithm's sensitivity to noise.
A must better weighting function, which substantially avoids this limitation is the gaussian function, which in python, looks like this:
def weight_gauss(dist, sig=2.0) :
return math.e**(-dist**2/(2*sig**2))
To calculate a predicted value using your kNN code, you would identify the n nearest neighbors to the data point whose response variable you wish to predict ('test instance'), then call the weight_gauss function, once for each of the n neighbors, passing in the distance between each neighbor the the test point.This function will return the weight for each neighbor, which is then used as that neighbor's coefficient in the weighted average calculation.
What you are facing is known as the curse of dimensionality. It is sometimes useful to run an algorithm like PCA or ICA to make sure that you really need all 21 dimensions and possibly find a linear transformation which would allow you to use less than 21 with approximately the same result quality.
Update:
I encountered them in a book called Biomedical Signal Processing by Rangayyan (I hope I remember it correctly). ICA is not a trivial technique, but it was developed by researchers in Finland and I think Matlab code for it is publicly available for download. PCA is a more widely used technique and I believe you should be able to find its R or other software implementation. PCA is performed by solving linear equations iteratively. I've done it too long ago to remember how. = )
The idea is that you break up your signals into independent eigenvectors (discrete eigenfunctions, really) and their eigenvalues, 21 in your case. Each eigenvalue shows the amount of contribution each eigenfunction provides to each of your measurements. If an eigenvalue is tiny, you can very closely represent the signals without using its corresponding eigenfunction at all, and that's how you get rid of a dimension.
Top answers are good but old, so I'd like to add up a 2016 answer.
As said, in a high dimensional space, the curse of dimensionality lurks around the corner, making the traditional approaches, such as the popular k-d tree, to be as slow as a brute force approach. As a result, we turn our interest in Approximate Nearest Neighbor Search (ANNS), which in favor of some accuracy, speedups the process. You get a good approximation of the exact NN, with a good propability.
Hot topics that might be worthy:
Modern approaches of LSH, such as Razenshteyn's.
RKD forest: Forest(s) of Randomized k-d trees (RKD), as described in FLANN,
or in a more recent approach I was part of, kd-GeRaF.
LOPQ which stands for Locally Optimized Product Quantization, as described here. It is very similar to the new Babenko+Lemptitsky's approach.
You can also check my relevant answers:
Two sets of high dimensional points: Find the nearest neighbour in the other set
Comparison of the runtime of Nearest Neighbor queries on different data structures
PCL kd-tree implementation extremely slow
To answer your questions one by one:
No, euclidean distance is a bad metric in high dimensional space. Basically in high dimensions, data points have large differences between each other. That decreases the relative difference in the distance between a given data point and its nearest and farthest neighbour.
Lot of papers/research are there in high dimension data, but most of the stuff requires a lot of mathematical sophistication.
KD tree is bad for high dimensional data ... avoid it by all means
Here is a nice paper to get you started in the right direction. "When in Nearest Neighbour meaningful?" by Beyer et all.
I work with text data of dimensions 20K and above. If you want some text related advice, I might be able to help you out.
Cosine similarity is a common way to compare high-dimension vectors. Note that since it's a similarity not a distance, you'd want to maximize it not minimize it. You can also use a domain-specific way to compare the data, for example if your data was DNA sequences, you could use a sequence similarity that takes into account probabilities of mutations, etc.
The number of nearest neighbors to use varies depending on the type of data, how much noise there is, etc. There are no general rules, you just have to find what works best for your specific data and problem by trying all values within a range. People have an intuitive understanding that the more data there is, the fewer neighbors you need. In a hypothetical situation where you have all possible data, you only need to look for the single nearest neighbor to classify.
The k Nearest Neighbor method is known to be computationally expensive. It's one of the main reasons people turn to other algorithms like support vector machines.
kd-trees indeed won't work very well on high-dimensional data. Because the pruning step no longer helps a lot, as the closest edge - a 1 dimensional deviation - will almost always be smaller than the full-dimensional deviation to the known nearest neighbors.
But furthermore, kd-trees only work well with Lp norms for all I know, and there is the distance concentration effect that makes distance based algorithms degrade with increasing dimensionality.
For further information, you may want to read up on the curse of dimensionality, and the various variants of it (there is more than one side to it!)
I'm not convinced there is a lot use to just blindly approximating Euclidean nearest neighbors e.g. using LSH or random projections. It may be necessary to use a much more fine tuned distance function in the first place!
A lot depends on why you want to know the nearest neighbors. You might look into the mean shift algorithm http://en.wikipedia.org/wiki/Mean-shift if what you really want is to find the modes of your data set.
I think cosine on tf-idf of boolean features would work well for most problems. That's because its time-proven heuristic used in many search engines like Lucene. Euclidean distance in my experience shows bad results for any text-like data. Selecting different weights and k-examples can be done with training data and brute-force parameter selection.
iDistance is probably the best for exact knn retrieval in high-dimensional data. You can view it as an approximate Voronoi tessalation.
I've experienced the same problem and can say the following.
Euclidean distance is a good distance metric, however it's computationally more expensive than the Manhattan distance, and sometimes yields slightly poorer results, thus, I'd choose the later.
The value of k can be found empirically. You can try different values and check the resulting ROC curves or some other precision/recall measure in order to find an acceptable value.
Both Euclidean and Manhattan distances respect the Triangle inequality, thus you can use them in metric trees. Indeed, KD-trees have their performance severely degraded when the data have more than 10 dimensions (I've experienced that problem myself). I found VP-trees to be a better option.
KD Trees work fine for 21 dimensions, if you quit early,
after looking at say 5 % of all the points.
FLANN does this (and other speedups)
to match 128-dim SIFT vectors. (Unfortunately FLANN does only the Euclidean metric,
and the fast and solid
scipy.spatial.cKDTree
does only Lp metrics;
these may or may not be adequate for your data.)
There is of course a speed-accuracy tradeoff here.
(If you could describe your Ndata, Nquery, data distribution,
that might help people to try similar data.)
Added 26 April, run times for cKDTree with cutoff on my old mac ppc, to give a very rough idea of feasibility:
kdstats.py p=2 dim=21 N=1000000 nask=1000 nnear=2 cutoff=1000 eps=0 leafsize=10 clustype=uniformp
14 sec to build KDtree of 1000000 points
kdtree: 1000 queries looked at av 0.1 % of the 1000000 points, 0.31 % of 188315 boxes; better 0.0042 0.014 0.1 %
3.5 sec to query 1000 points
distances to 2 nearest: av 0.131 max 0.253
kdstats.py p=2 dim=21 N=1000000 nask=1000 nnear=2 cutoff=5000 eps=0 leafsize=10 clustype=uniformp
14 sec to build KDtree of 1000000 points
kdtree: 1000 queries looked at av 0.48 % of the 1000000 points, 1.1 % of 188315 boxes; better 0.0071 0.026 0.5 %
15 sec to query 1000 points
distances to 2 nearest: av 0.131 max 0.245
You could try a z order curve. It's easy for 3 dimension.
I had a similar question a while back. For fast Approximate Nearest Neighbor Search you can use the annoy library from spotify: https://github.com/spotify/annoy
This is some example code for the Python API, which is optimized in C++.
from annoy import AnnoyIndex
import random
f = 40
t = AnnoyIndex(f, 'angular') # Length of item vector that will be indexed
for i in range(1000):
v = [random.gauss(0, 1) for z in range(f)]
t.add_item(i, v)
t.build(10) # 10 trees
t.save('test.ann')
# ...
u = AnnoyIndex(f, 'angular')
u.load('test.ann') # super fast, will just mmap the file
print(u.get_nns_by_item(0, 1000)) # will find the 1000 nearest neighbors
They provide different distance measurements. Which distance measurement you want to apply depends highly on your individual problem. Also consider prescaling (meaning weighting) certain dimensions for importance first. Those dimension or feature importance weights might be calculated by something like entropy loss or if you have a supervised learning problem gini impurity gain or mean average loss, where you check how much worse your machine learning model performs, if you scramble this dimensions values.
Often the direction of the vector is more important than it's absolute value. For example in the semantic analysis of text documents, where we want document vectors to be close when their semantics are similar, not their lengths. Thus we can either normalize those vectors to unit length or use angular distance (i.e. cosine similarity) as a distance measurement.
Hope this is helpful.
Is Euclidean distance a good metric for finding the nearest neighbors in the first place? If not, what are my options?
I would suggest soft subspace clustering, a pretty common approach nowadays, where feature weights are calculated to find the most relevant dimensions. You can use these weights when using euclidean distance, for example. See curse of dimensionality for common problems and also this article can enlighten you somehow:
A k-means type clustering algorithm for subspace clustering of mixed numeric and
categorical datasets
I have need to do some cluster analysis on a set of 2 dimensional data (I may add extra dimensions along the way).
The analysis itself will form part of the data being fed into a visualisation, rather than the inputs into another process (e.g. Radial Basis Function Networks).
To this end, I'd like to find a set of clusters which primarily "looks right", rather than elucidating some hidden patterns.
My intuition is that k-means would be a good starting place for this, but that finding the right number of clusters to run the algorithm with would be problematic.
The problem I'm coming to is this:
How to determine the 'best' value for k such that the clusters formed are stable and visually verifiable?
Questions:
Assuming that this isn't NP-complete, what is the time complexity for finding a good k. (probably reported in number of times to run the k-means algorithm).
is k-means a good starting point for this type of problem? If so, what other approaches would you recommend. A specific example, backed by an anecdote/experience would be maxi-bon.
what short cuts/approximations would you recommend to increase the performance.
For problems with an unknown number of clusters, agglomerative hierarchical clustering is often a better route than k-means.
Agglomerative clustering produces a tree structure, where the closer you are to the trunk, the fewer the number of clusters, so it's easy to scan through all numbers of clusters. The algorithm starts by assigning each point to its own cluster, and then repeatedly groups the two closest centroids. Keeping track of the grouping sequence allows an instant snapshot for any number of possible clusters. Therefore, it's often preferable to use this technique over k-means when you don't know how many groups you'll want.
There are other hierarchical clustering methods (see the paper suggested in Imran's comments). The primary advantage of an agglomerative approach is that there are many implementations out there, ready-made for your use.
In order to use k-means, you should know how many cluster there is. You can't try a naive meta-optimisation, since the more cluster you'll add (up to 1 cluster for each data point), the more it will brought you to over-fitting. You may look for some cluster validation methods and optimize the k hyperparameter with it but from my experience, it rarely work well. It's very costly too.
If I were you, I would do a PCA, eventually on polynomial space (take care of your available time) depending on what you know of your input, and cluster along the most representatives components.
More infos on your data set would be very helpful for a more precise answer.
Here's my approximate solution:
Start with k=2.
For a number of tries:
Run the k-means algorithm to find k clusters.
Find the mean square distance from the origin to the cluster centroids.
Repeat the 2-3, to find a standard deviation of the distances. This is a proxy for the stability of the clusters.
If stability of clusters for k < stability of clusters for k - 1 then return k - 1
Increment k by 1.
The thesis behind this algorithm is that the number of sets of k clusters is small for "good" values of k.
If we can find a local optimum for this stability, or an optimal delta for the stability, then we can find a good set of clusters which cannot be improved by adding more clusters.
In a previous answer, I explained how Self-Organizing Maps (SOM) can be used in visual clustering.
Otherwise, there exist a variation of the K-Means algorithm called X-Means which is able to find the number of clusters by optimizing the Bayesian Information Criterion (BIC), in addition to solving the problem of scalability by using KD-trees.
Weka includes an implementation of X-Means along with many other clustering algorithm, all in an easy to use GUI tool.
Finally you might to refer to this page which discusses the Elbow Method among other techniques for determining the number of clusters in a dataset.
You might look at papers on cluster validation. Here's one that is cited in papers that involve microarray analysis, which involves clustering genes with related expression levels.
One such technique is the Silhouette measure that evaluates how closely a labeled point is to its centroid. The general idea is that, if a point is assigned to one centroid but is still close to others, perhaps it was assigned to the wrong centroid. By counting these events across training sets and looking across various k-means clusterings, one looks for the k such that the labeled points overall fall into the "best" or minimally ambiguous arrangement.
It should be said that clustering is more of a data visualization and exploration technique. It can be difficult to elucidate with certainty that one clustering explains the data correctly, above all others. It's best to merge your clusterings with other relevant information. Is there something functional or otherwise informative about your data, such that you know some clusterings are impossible? This can reduce your solution space considerably.
From your wikipedia link:
Regarding computational complexity,
the k-means clustering problem is:
NP-hard in general Euclidean
space d even for 2 clusters
NP-hard for a general number of
clusters k even in the plane
If k and d are fixed, the problem can be
exactly solved in time O(ndk+1 log n),
where n is the number of entities to
be clustered
Thus, a variety of heuristic
algorithms are generally used.
That said, finding a good value of k is usually a heuristic process (i.e. you try a few and select the best).
I think k-means is a good starting point, it is simple and easy to implement (or copy). Only look further if you have serious performance problems.
If the set of points you want to cluster is exceptionally large a first order optimisation would be to randomly select a small subset, use that set to find your k-means.
Choosing the best K can be seen as a Model Selection problem. One possible approach is Minimum Description Length, which in this context means: You could store a table with all the points (in which case K=N). At the other extreme, you have K=1, and all the points are stored as their distances from a single centroid. This Section from Introduction to Information Retrieval by Manning and Schutze suggest minimising the Akaike Information Criterion as a heuristic for an optimal K.
This problematic belongs to the "internal evaluation" class of "clustering optimisation problems" which curent state of the art solution seems to use the **Silhouette* coeficient* as stated here
https://en.wikipedia.org/wiki/Cluster_analysis#Applications
and here:
https://en.wikipedia.org/wiki/Silhouette_(clustering) :
"silhouette plots and averages may be used to determine the natural number of clusters within a dataset"
scikit-learn provides a sample usage implementation of the methodology here
http://scikit-learn.org/stable/auto_examples/cluster/plot_kmeans_silhouette_analysis.html