Find correlation in large dataset - algorithm

I have a huge dataset. We are talking about 100 3D matrices with 121x145x121 cells. Any cell has a value between 0 and 1, and I need a way to cluster these cells according to their correlation. The problem is the dataset is too big for any algorithm I know; even using just half of it (any matrix is a MRI scan of a brain) we have around 400 billion pairs. Any ideas?

As a first step I would be tempted to try K-means clustering.
This appears in the Matlab statistics toolbox as the function kmeans.
In this algorithm you only end up computing the distances between the K current centres and the data, so the number of pairs is much smaller than comparing all choices.
In Matlab, I've also found that the speed of the operation can be quite dependent on the organisation of your matrix (due to memory caching and optimisation issues). I would recommend transforming your 3d matrices so that the columns (held together in memory) correspond to the 100 values for a particular cell.
This can be done with the permute function.

Try a weighted K-means++ clustering algorithm. Create one matrix of the sum of values for all the 100 input matrices at every point to produce one "grey scale" matrix, then adjust the K-means++ algorithm to work with weighted, (wt), values.
In the initialization phase choose one new data point at random as a new center, using a weighted probability distribution where a point x is chosen with probability proportional to D(X)^2 x wt^2 .
The assignment step should be okay, but when computing the centroids in the update step adjust the formula to account for the weights. (Or use the same formula but each point is used wt times).
You may not be able to use a library function to do this but you start with a 100 fold decrease in number of points and matrices to work with.

Related

Divide 2D array into continuous regions of as-equal-as-possible sums?

I have a 2D array of floating-point numbers, and I'd like to divide this array into an arbitrary number of regions such that the sum of all the regions' elements are more or less equal. The regions must be continuous. By as-equal-as-possible, I mean that the standard deviation of the region sums should be reduced as much as possible.
I'm doing this because I have a map of values corresponding to the "population" in an area, and I want to divide this area into groups of relatively equal population.
Thanks!
I would do it like this:
1.compute the whole sum
2.compute local centers of mass (coordinates)
3.now compute the region sum
for example:
region sum = whole sum / number of centers of masses
4.for each center of mass
start a region
and incrementally increase the size until it sum match region sum
avoid intersection of regions (use some map of usage for that)
if region has the desired sum or has nowhere to grow stop
You will have to tweak this algorithm a little to suite your needs and input data
Hope it helps a little ...
Standard deviation is way to measure that whether the divisions are close to equal. Lower standard deviation means closer the sums are.
As the problem seems n-p like clustering problems , Genetic algorithms can be used to get good solutions to the problem :-
Standard deviation can be used as fitness measure for chromosomes.
Consider k contagious regions then each gene(element) will have one of the k values which maintain the contagious nature of the regions.
apply genetic algorithm on the chromosomes and get the best chromosome for that value of k after a fixed amount of generations.
vary k from 2 to n and get best chromosome by applying genetic algorithms.

"Covering" the space of all possible histogram shapes

There is a very expensive computation I must make frequently.
The computation takes a small array of numbers (with about 20 entries) that sums to 1 (i.e. the histogram) and outputs something that I can store pretty easily.
I have 2 things going for me:
I can accept approximate answers
The "answers" change slowly. For example: [.1 .1 .8 0] and [.1
.1 .75 .05] will yield similar results.
Consequently, I want to build a look-up table of answers off-line. Then, when the system is running, I can look-up an approximate answer based on the "shape" of the input histogram.
To be precise, I plan to look-up the precomputed answer that corresponds to the histogram with the minimum Earth-Mover-Distance to the actual input histogram.
I can only afford to store about 80 to 100 precomputed (histogram , computation result) pairs in my look up table.
So, how do I "spread out" my precomputed histograms so that, no matter what the input histogram is, I'll always have a precomputed result that is "close"?
Finding N points in M-space that are a best spread-out set is more-or-less equivalent to hypersphere packing (1,2) and in general answers are not known for M>10. While a fair amount of research has been done to develop faster methods for hypersphere packings or approximations, it is still regarded as a hard problem.
It probably would be better to apply a technique like principal component analysis or factor analysis to as large a set of histograms as you can conveniently generate. The results of either analysis will be a set of M numbers such that linear combinations of histogram data elements weighted by those numbers will predict some objective function. That function could be the “something that you can store pretty easily” numbers, or could be case numbers. Also consider developing and training a neural net or using other predictive modeling techniques to predict the objective function.
Building on #jwpat7's answer, I would apply k-means clustering to a huge set of randomly generated (and hopefully representative) histograms. This would ensure that your space was spanned with whatever number of exemplars (precomputed results) you can support, with roughly equal weighting for each cluster.
The trick, of course, will be generating representative data to cluster in the first place. If you can recompute from time to time, you can recluster based on the actual data in the system so that your clusters might get better over time.
I second jwpat7's answer, but my very naive approach was to consider the count of items in each histogram bin as a y value, to consider the x values as just 0..1 in 20 steps, and then to obtain parameters a,b,c that describe x vs y as a cubic function.
To get a "covering" of the histograms I just iterated through "possible" values for each parameter.
e.g. to get 27 histograms to cover the "shape space" of my cubic histogram model I iterated the parameters through -1 .. 1, choosing 3 values linearly spaced.
Now, you could change the histogram model to be quartic if you think your data will often be represented that way, or whatever model you think is most descriptive, as well as generate however many histograms to cover. I used 27 because three partitions per parameter for three parameters is 3*3*3=27.
For a more comprehensive covering, like 100, you would have to more carefully choose your ranges for each parameter. 100**.3 isn't an integer, so the simple num_covers**(1/num_params) solution wouldn't work, but for 3 parameters 4*5*5 would.
Since the actual values of the parameters could vary greatly and still achieve the same shape it would probably be best to store ratios of them for comparison instead, e.g. for my 3 parmeters b/a and b/c.
Here is an 81 histogram "covering" using a quartic model, again with parameters chosen from linspace(-1,1,3):
edit: Since you said your histograms were described by arrays that were ~20 elements, I figured fitting parameters would be very fast.
edit2 on second thought I think using a constant in the model is pointless, all that matters is the shape.

Single Pass Seed Selection Algorithm for k-Means

I've recently read the Single Pass Seed Selection Algorithm for k-Means article, but not really understand the algorithm, which is:
Calculate distance matrix Dist in which Dist (i,j) represents distance from i to j
Find Sumv in which Sumv (i) is the sum of the distances from ith point to all other points.
Find the point i which is min (Sumv) and set Index = i
Add First to C as the first centroid
For each point xi, set D (xi) to be the distance between xi and the nearest point in C
Find y as the sum of distances of first n/k nearest points from the Index
Find the unique integer i so that D(x1)^2+D(x2)^2+...+D(xi)^2 >= y > D(x1)^2+D(x2)^2+...+D(x(i-1))^2
Add xi to C
Repeat steps 5-8 until k centers
Especially step 6, do we still use the same Index (same point) over and over or we use the newly added point from C? And about step 8, does i have to be larger than 1?
Honestly, I wouldn't worry about understanding that paper - its not very good.
The algorithm is poorly described.
Its not actually a single pass, it needs do to n^2/2 pairwise computations + one additional pass through the data.
They don't report the runtime of their seed selection scheme, probably because it is very bad doing O(n^2) work.
They are evaluating on very simple data sets that don't have a lot of bad solutions for k-Means to fall into.
One of their metrics of "better"ness is how many iterations it takes k-means to run given the seed selection. While it is an interesting metric, the small differences they report are meaningless (k-means++ seeding could be more iterations, but less work done per iteration), and they don't report the run time or which k-means algorithm they use.
You will get a lot more benefit from learning and understanding the k-means++ algorithm they are comparing against, and reading some of the history from that.
If you really want to understand what they are doing, I would brush up on your matlab and read their provided matlab code. But its not really worth it. If you look up the quantile seed selection algorithm, they are essentially doing something very similar. Instead of using the distance to the first seed to sort the points, they appear to be using the sum of pairwise distances (which means they don't need an initial seed, hence the unique solution).
Single Pass Seed Selection algorithm is a novel algorithm. Single Pass mean that without any iterations first seed can be selected. k-means++ performance is depends on first seed. It is overcome in SPSS. Please gothrough the paper "Robust Seed Selestion Algorithm for k-means" from the same authors
John J. Louis

Random projection algorithm pseudo code

I am trying to apply Random Projections method on a very sparse dataset. I found papers and tutorials about Johnson Lindenstrauss method, but every one of them is full of equations which makes no meaningful explanation to me. For example, this document on Johnson-Lindenstrauss
Unfortunately, from this document, I can get no idea about the implementation steps of the algorithm. It's a long shot but is there anyone who can tell me the plain English version or very simple pseudo code of the algorithm? Or where can I start to dig this equations? Any suggestions?
For example, what I understand from the algorithm by reading this paper concerning Johnson-Lindenstrauss is that:
Assume we have a AxB matrix where A is number of samples and B is the number of dimensions, e.g. 100x5000. And I want to reduce the dimension of it to 500, which will produce a 100x500 matrix.
As far as I understand: first, I need to construct a 100x500 matrix and fill the entries randomly with +1 and -1 (with a 50% probability).
Edit:
Okay, I think I started to get it. So we have a matrix A which is mxn. We want to reduce it to E which is mxk.
What we need to do is, to construct a matrix R which has nxk dimension, and fill it with 0, -1 or +1, with respect to 2/3, 1/6 and 1/6 probability.
After constructing this R, we'll simply do a matrix multiplication AxR to find our reduced matrix E. But we don't need to do a full matrix multiplication, because if an element of Ri is 0, we don't need to do calculation. Simply skip it. But if we face with 1, we just add the column, or if it's -1, just subtract it from the calculation. So we'll simply use summation rather than multiplication to find E. And that is what makes this method very fast.
It turned out a very neat algorithm, although I feel too stupid to get the idea.
You have the idea right. However as I understand random project, the rows of your matrix R should have unit length. I believe that's approximately what the normalizing by 1/sqrt(k) is for, to normalize away the fact that they're not unit vectors.
It isn't a projection, but, it's nearly a projection; R's rows aren't orthonormal, but within a much higher-dimensional space, they quite nearly are. In fact the dot product of any two of those vectors you choose will be pretty close to 0. This is why it is a generally good approximation of actually finding a proper basis for projection.
The mapping from high-dimensional data A to low-dimensional data E is given in the statement of theorem 1.1 in the latter paper - it is simply a scalar multiplication followed by a matrix multiplication. The data vectors are the rows of the matrices A and E. As the author points out in section 7.1, you don't need to use a full matrix multiplication algorithm.
If your dataset is sparse, then sparse random projections will not work well.
You have a few options here:
Option A:
Step 1. apply a structured dense random projection (so called fast hadamard transform is typically used). This is a special projection which is very fast to compute but otherwise has the properties of a normal dense random projection
Step 2. apply sparse projection on the "densified data" (sparse random projections are useful for dense data only)
Option B:
Apply SVD on the sparse data. If the data is sparse but has some structure SVD is better. Random projection preserves the distances between all points. SVD preserves better the distances between dense regions - in practice this is more meaningful. Also people use random projections to compute the SVD on huge datasets. Random Projections gives you efficiency, but not necessarily the best quality of embedding in a low dimension.
If your data has no structure, then use random projections.
Option C:
For data points for which SVD has little error, use SVD; for the rest of the points use Random Projection
Option D:
Use a random projection based on the data points themselves.
This is very easy to understand what is going on. It looks something like this:
create a n by k matrix (n number of data point, k new dimension)
for i from 0 to k do #generate k random projection vectors
randomized_combination = feature vector of zeros (number of zeros = number of features)
sample_point_ids = select a sample of point ids
for each point_id in sample_point_ids do:
random_sign = +1/-1 with prob. 1/2
randomized_combination += random_sign*feature_vector[point_id] #this is a vector operation
normalize the randomized combination
#note that the normal random projection is:
# randomized_combination = [+/-1, +/-1, ...] (k +/-1; if you want sparse randomly set a fraction to 0; also good to normalize by length]
to project the data points on this random feature just do
for each data point_id in dataset:
scores[point_id, j] = dot_product(feature_vector[point_id], randomized_feature)
If you are still looking to solve this problem, write a message here, I can give you more pseudocode.
The way to think about it is that a random projection is just a random pattern and the dot product (i.e. projecting the data point) between the data point and the pattern gives you the overlap between them. So if two data points overlap with many random patterns, those points are similar. Therefore, random projections preserve similarity while using less space, but they also add random fluctuations in the pairwise similarities. What JLT tells you is that to make fluctuations 0.1 (eps)
you need about 100*log(n) dimensions.
Good Luck!
An R Package to perform Random Projection using Johnson- Lindenstrauss Lemma
RandPro

Nearest neighbors in high-dimensional data?

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

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