I apologize if my questions are extremely misguided or loosely scoped. Math is not my strongest subject. For context, I am trying to figure out the computational complexity of calculating the area under a discrete curve. In the particular use case that I am interested in, the y-axis is the length of a queue and the x-axis is time. The curve will always have the following bounds: it begins at zero, it is composed of multiple timestamped samples that are greater than zero, and it eventually shrinks to zero. My initial research has yielded two potential mathematical approaches to this problem. The first is a Reimann sum over domain [a, b] where a is initially zero and b eventually becomes zero (not sure if my understanding is completely correct there). I think the mathematical representation of this the formula found here:
https://en.wikipedia.org/wiki/Riemann_sum#Connection_with_integration.
The second is a discrete convolution. However, I am unable to tell the difference between, and applicability of, a discrete convolution and a Reimann sum over domain [a, b] where a is initially zero and b eventually becomes zero.
My questions are:
Is there are difference between the two?
Which approach is most applicable/efficient for what I am trying to figure out?
Is it even appropriate ask the computation complexity of either mathematical approach? If so, what are the complexities of each in this particular application?
Edit:
For added context, there will be a function calculating average queue length by taking the sum of the area under two separate curves and dividing it by the total time interval spanning those two curves. The particular application can be seen on page 168 of this paper: https://www.cse.wustl.edu/~jain/cv/raj_jain_paper4_decbit.pdf
Is there are difference between the two?
A discrete convolution requires two functions. If the first one corresponds to the discrete curve, what is the second one?
Which approach is most applicable/efficient for what I am trying to figure out?
A Riemann sum is an approximation of an integral. It's typically used to approximate the area under a continuous curve. You can of course use it on a discrete curve, but it's not an approximation anymore, and I'm not sure you can call it a "Riemann" sum.
Is it even appropriate ask the computation complexity of either mathematical approach? If so, what are the complexities of each in this particular application?
In any case, the complexity of computing the area under a dicrete curve is linear in the number of samples, and it's pretty straightforward to find why: you need to do something with each sample, once or twice.
What you probably want looks like a Riemann sum with the trapezoidal rule. Pick the first two samples, calculate their average, and multiply that by the distance between two samples. Repeat for every adjacent pair and sum it all.
So, this is for the router feedback filter in the referenced paper...
That algorithm is specifically designed so that you can implement it without storing a lot of samples and timestamps.
It works by accumulating total queue_length * time during each cycle.
At the start of each "cycle", record the current queue length and current clock time and set the current cycle's total to 0. (The paper defines the cycle so that the queue length is 0 at the start, but that's not important here)
every time the queue length changes, get the new current clock time and add (new_clock_time - previous_clock_time) * previous_queue_length to the total. Also do this at the end of the cycle. Then, record new new current queue length and current clock time.
When you need to calculate the current "average queue length", it's just (previous_cycle_total + current_cycle_total + (current_clock_time - previous_clock_time)*previous_queue_length) / total_time_since_previous_cycle_start
I'm working on a problem that necessitates running KMeans separately on ~125 different datasets. Therefore, I'm looking to mathematically calculate the 'optimal' K for each respective dataset. However, the evaluation metric continues decreasing with higher K values.
For a sample dataset, there are 50K rows and 8 columns. Using sklearn's calinski-harabaz score, I'm iterating through different K values to find the optimum / minimum score. However, my code reached k=5,600 and the calinski-harabaz score was still decreasing!
Something weird seems to be happening. Does the metric not work well? Could my data be flawed (see my question about normalizing rows after PCA)? Is there another/better way to mathematically converge on the 'optimal' K? Or should I force myself to manually pick a constant K across all datasets?
Any additional perspectives would be helpful. Thanks!
I don't know anything about the calinski-harabaz score but some score metrics will be monotone increasing/decreasing with respect to increasing K. For instance the mean squared error for linear regression will always decrease each time a new feature is added to the model so other scores that add penalties for increasing number of features have been developed.
There is a very good answer here that covers CH scores well. A simple method that generally works well for these monotone scoring metrics is to plot K vs the score and choose the K where the score is no longer improving 'much'. This is very subjective but can still give good results.
SUMMARY
The metric decreases with each increase of K; this strongly suggests that you do not have a natural clustering upon the data set.
DISCUSSION
CH scores depend on the ratio between intra- and inter-cluster densities. For a relatively smooth distribution of points, each increase in K will give you clusters that are slightly more dense, with slightly lower density between them. Try a lattice of points: vary the radius and do the computations by hand; you'll see how that works. At the extreme end, K = n: each point is its own cluster, with infinite density, and 0 density between clusters.
OTHER METRICS
Perhaps the simplest metric is sum-of-squares, which is already part of the clustering computations. Sum the squares of distances from the centroid, divide by n-1 (n=cluster population), and then add/average those over all clusters.
I'm looking for a particular paper that discusses metrics for this very problem; if I can find the reference, I'll update this answer.
N.B. With any metric you choose (as with CH), a failure to find a local minimum suggests that the data really don't have a natural clustering.
WHAT TO DO NEXT?
Render your data in some form you can visualize. If you see a natural clustering, look at the characteristics; how is it that you can see it, but the algebra (metrics) cannot? Formulate a metric that highlights the differences you perceive.
I know, this is an effort similar to the problem you're trying to automate. Welcome to research. :-)
The problem with my question is that the 'best' Calinski-Harabaz score is the maximum, whereas my question assumed the 'best' was the minimum. It is computed by analyzing the ratio of between-cluster dispersion vs. within-cluster dispersion, the former/numerator you want to maximize, the latter/denominator you want to minimize. As it turned out, in this dataset, the 'best' CH score was with 2 clusters (the minimum available for comparison). I actually ran with K=1, and this produced good results as well. As Prune suggested, there appears to be no natural grouping within the dataset.
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'm observing a sinusoidally-varying source, i.e. f(x) = a sin (bx + d) + c, and want to determine the amplitude a, offset c and period/frequency b - the shift d is unimportant. Measurements are sparse, with each source measured typically between 6 and 12 times, and observations are at (effectively) random times, with intervals between observations roughly between a quarter and ten times the period (just to stress, the spacing of observations is not constant for each source). In each source the offset c is typically quite large compared to the measurement error, while amplitudes vary - at one extreme they are only on the order of the measurement error, while at the other extreme they are about twenty times the error. Hopefully that fully outlines the problem, if not, please ask and i'll clarify.
Thinking naively about the problem, the average of the measurements will be a good estimate of the offset c, while half the range between the minimum and maximum value of the measured f(x) will be a reasonable estimate of the amplitude, especially as the number of measurements increase so that the prospects of having observed the maximum offset from the mean improve. However, if the amplitude is small then it seems to me that there is little chance of accurately determining b, while the prospects should be better for large-amplitude sources even if they are only observed the minimum number of times.
Anyway, I wrote some code to do a least-squares fit to the data for the range of periods, and it identifies best-fit values of a, b and d quite effectively for the larger-amplitude sources. However, I see it finding a number of possible periods, and while one is the 'best' (in as much as it gives the minimum error-weighted residual) in the majority of cases the difference in the residuals for different candidate periods is not large. So what I would like to do now is quantify the possibility that the derived period is a 'false positive' (or, to put it slightly differently, what confidence I can have that the derived period is correct).
Does anybody have any suggestions on how best to proceed? One thought I had was to use a Monte-Carlo algorithm to construct a large number of sources with known values for a, b and c, construct samples that correspond to my measurement times, fit the resultant sample with my fitting code, and see what percentage of the time I recover the correct period. But that seems quite heavyweight, and i'm not sure that it's particularly useful other than giving a general feel for the false-positive rate.
And any advice for frameworks that might help? I have a feeling this is something that can likely be done in a line or two in Mathematica, but (a) I don't know it, an (b) don't have access to it. I'm fluent in Java, competent in IDL and can probably figure out other things...
This looks tailor-made for working in the frequency domain. Apply a Fourier transform and identify the frequency based on where the power is located, which should be clear for a sinusoidal source.
ADDENDUM To get an idea of how accurate is your estimate, I'd try a resampling approach such as cross-validation. I think this is the direction that you're heading with the Monte Carlo idea; lots of work is out there, so hopefully that's a wheel you won't need to re-invent.
The trick here is to do what might seem at first to make the problem more difficult. Rewrite f in the similar form:
f(x) = a1*sin(b*x) + a2*cos(b*x) + c
This is based on the identity for the sin(u+v).
Recognize that if b is known, then the problem of estimating {a1, a2, c} is a simple LINEAR regression problem. So all you need to do is use a 1-variable minimization tool, working on the value of b, to minimize the sum of squares of the residuals from that linear regression model. There are many such univariate optimizers to be found.
Once you have those parameters, it is easy to find the parameter a in your original model, since that is all you care about.
a = sqrt(a1^2 + a2^2)
The scheme I have described is called a partitioned least squares.
If you have a reasonable estimate of the size and the nature of your noise (e.g. white Gaussian with SD sigma), you can
(a) invert the Hessian matrix to get an estimate of the error in your position and
(b) should be able to easily derive a significance statistic for your fit residues.
For (a), compare http://www.physics.utah.edu/~detar/phys6720/handouts/curve_fit/curve_fit/node6.html
For (b), assume that your measurement errors are independent and thus the variance of their sum is the sum of their variances.
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