Find k-nearest neighbors using searcher object
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Syntax
Idx = knnsearch(Mdl,Y)
Idx = knnsearch(Mdl,Y,Name,Value)
[Idx,D]= knnsearch(___)
Description
example
Idx = knnsearch(Mdl,Y) searches for the nearest neighbor (i.e., the closest point, row, or observation) in Mdl.X to each point (i.e., row or observation) in the query data Y using an exhaustive search, a Kd-tree, or a Hierarchical Navigable Small Worlds approximate search. knnsearch returns Idx, which is a column vector of the indices in Mdl.X representing the nearest neighbors.
example
Idx = knnsearch(Mdl,Y,Name,Value) returns the indices of the closest points in Mdl.X to Y with additional options specified by one or more Name,Value pair arguments. For example, specify the number of nearest neighbors to search for, or distance metric different from the one stored in Mdl.Distance. You can also specify which action to take if the closest distances are tied.
example
[Idx,D]= knnsearch(___) additionally returns the matrix D using any of the input arguments in the previous syntaxes. D contains the distances between each observation in Y that correspond to the closest observations in Mdl.X. By default, the function arranges the columns of D in ascending order by closeness, with respect to the distance metric.
Examples
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Search for Nearest Neighbors Using Kd-tree and Exhaustive Search
Open Live Script
knnsearch accepts ExhaustiveSearcher or KDTreeSearcher model objects to search the training data for the nearest neighbors to the query data. An ExhaustiveSearcher model invokes the exhaustive searcher algorithm, and a KDTreeSearcher model defines a Kd-tree, which knnsearch uses to search for nearest neighbors.
Load Fisher's iris data set. Randomly reserve five observations from the data for query data.
load fisheririsrng(1); % For reproducibilityn = size(meas,1);idx = randsample(n,5);X = meas(~ismember(1:n,idx),:); % Training dataY = meas(idx,:); % Query data
The variable meas contains 4 predictors.
Grow a default four-dimensional Kd-tree.
MdlKDT = KDTreeSearcher(X)
MdlKDT = KDTreeSearcher with properties: BucketSize: 50 Distance: 'euclidean' DistParameter: [] X: [145x4 double]
MdlKDT is a KDTreeSearcher model object. You can alter its writable properties using dot notation.
Prepare an exhaustive nearest neighbor searcher.
MdlES = ExhaustiveSearcher(X)
MdlES = ExhaustiveSearcher with properties: Distance: 'euclidean' DistParameter: [] X: [145x4 double]
MdlKDT is an ExhaustiveSearcher model object. It contains the options, such as the distance metric, to use to find nearest neighbors.
Alternatively, you can grow a Kd-tree or prepare an exhaustive nearest neighbor searcher using createns.
Search the training data for the nearest neighbors indices that correspond to each query observation. Conduct both types of searches using the default settings. By default, the number of neighbors to search for per query observation is 1.
IdxKDT = knnsearch(MdlKDT,Y);IdxES = knnsearch(MdlES,Y);[IdxKDT IdxES]
ans = 5×2 17 17 6 6 1 1 89 89 124 124In this case, the results of the search are the same.
Search for Nearest Neighbors of Query Data Using Minkowski Distance
Open Live Script
Grow a Kd-tree nearest neighbor searcher object by using the createns function. Pass the object and query data to the knnsearch function to find k-nearest neighbors.
Load Fisher's iris data set.
load fisheririsRemove five irises randomly from the predictor data to use as a query set.
rng(1) % For reproducibilityn = size(meas,1); % Sample sizeqIdx = randsample(n,5); % Indices of query datatIdx = ~ismember(1:n,qIdx); % Indices of training dataQ = meas(qIdx,:);X = meas(tIdx,:);
Grow a four-dimensional Kd-tree using the training data. Specify the Minkowski distance for finding nearest neighbors.
Mdl = createns(X,'Distance','minkowski')
Mdl = KDTreeSearcher with properties: BucketSize: 50 Distance: 'minkowski' DistParameter: 2 X: [145x4 double]
Because X has four columns and the distance metric is Minkowski, createns creates a KDTreeSearcher model object by default. The Minkowski distance exponent is 2 by default.
Find the indices of the training data (Mdl.X) that are the two nearest neighbors of each point in the query data (Q).
IdxNN = knnsearch(Mdl,Q,'K',2)IdxNN = 5×2 17 4 6 2 1 12 89 66 124 100Each row of IdxNN corresponds to a query data observation, and the column order corresponds to the order of the nearest neighbors, with respect to ascending distance. For example, based on the Minkowski distance, the second nearest neighbor of Q(3,:) is X(12,:).
Include Ties in Nearest Neighbor Search
Open Live Script
Load Fisher's iris data set.
Remove five irises randomly from the predictor data to use as a query set.
rng(4); % For reproducibilityn = size(meas,1); % Sample sizeqIdx = randsample(n,5); % Indices of query dataX = meas(~ismember(1:n,qIdx),:);Y = meas(qIdx,:);
Grow a four-dimensional Kd-tree using the training data. Specify the Minkowski distance for finding nearest neighbors.
Mdl = KDTreeSearcher(X);
Mdl is a KDTreeSearcher model object. By default, the distance metric for finding nearest neighbors is the Euclidean metric.
Find the indices of the training data (X) that are the seven nearest neighbors of each point in the query data (Y).
[Idx,D] = knnsearch(Mdl,Y,'K',7,'IncludeTies',true);
Idx and D are five-element cell arrays of vectors, with each vector having at least seven elements.
Display the lengths of the vectors in Idx.
cellfun('length',Idx)ans = 5×1 8 7 7 7 7Because cell 1 contains a vector with length greater than k = 7, query observation 1 (Y(1,:)) is equally close to at least two observations in X.
Display the indices of the nearest neighbors to Y(1,:) and their distances.
nn5 = Idx{1}nn5 = 1×8 91 98 67 69 71 93 88 95nn5d = D{1}nn5d = 1×8 0.1414 0.2646 0.2828 0.3000 0.3464 0.3742 0.3873 0.3873Training observations 88 and 95 are 0.3873 cm away from query observation 1.
Compare k-Nearest Neighbors Using Different Distance Metrics
Open Live Script
Train two KDTreeSearcher models using different distance metrics, and compare k-nearest neighbors of query data for the two models.
Load Fisher's iris data set. Consider the petal measurements as predictors.
load fisheririsX = meas(:,3:4); % PredictorsY = species; % Response
Train a KDTreeSearcher model object by using the predictors. Specify the Minkowski distance with exponent 5.
KDTreeMdl = KDTreeSearcher(X,'Distance','minkowski','P',5)
KDTreeMdl = KDTreeSearcher with properties: BucketSize: 50 Distance: 'minkowski' DistParameter: 5 X: [150x2 double]
Find the 10 nearest neighbors from X to a query point (newpoint), first using Minkowski then Chebychev distance metrics. The query point must have the same column dimension as the data used to train the model.
newpoint = [5 1.45];[IdxMk,DMk] = knnsearch(KDTreeMdl,newpoint,'k',10);[IdxCb,DCb] = knnsearch(KDTreeMdl,newpoint,'k',10,'Distance','chebychev');
IdxMk and IdxCb are 1-by-10 matrices containing the row indices of X corresponding to the nearest neighbors to newpoint using Minkowski and Chebychev distances, respectively. Element (1,1) is the nearest, element (1,2) is the next nearest, and so on.
Plot the training data, query point, and nearest neighbors.
figure;gscatter(X(:,1),X(:,2),Y);title('Fisher''s Iris Data -- Nearest Neighbors');xlabel('Petal length (cm)');ylabel('Petal width (cm)');hold onplot(newpoint(1),newpoint(2),'kx','MarkerSize',10,'LineWidth',2); % Query point plot(X(IdxMk,1),X(IdxMk,2),'o','Color',[.5 .5 .5],'MarkerSize',10); % Minkowski nearest neighborsplot(X(IdxCb,1),X(IdxCb,2),'p','Color',[.5 .5 .5],'MarkerSize',10); % Chebychev nearest neighborslegend('setosa','versicolor','virginica','query point',... 'minkowski','chebychev','Location','Best');
Zoom in on the points of interest.
h = gca; % Get current axis handle.h.XLim = [4.5 5.5];h.YLim = [1 2];axis square;
Several observations are equal, which is why only eight nearest neighbors are identified in the plot.
HNSW Speed and Accuracy for Large Data
Open Live Script
Create a large data set and compare the speed of an HNSW searcher to the speed of an exhaustive searcher for 1000 query points.
Create the data.
rng default % For reproducibilityA = diag(1:100);B = randn(100,10);K = kron(A,B);ss = size(K)
ss = 1×2 10000 1000The data K has 1e4 rows and 1e3 columns.
Create an HNSW searcher object h from the data K.
tic;h = hnswSearcher(K);chnsw = toc
chnsw = 10.6544
Create 1000 random query points with 1000 features (columns). Specify to search for five nearest neighbors.
Y = randn(1000, 1000);tic;[idx, D] = knnsearch(h,Y,K=5);thnsw = toc
thnsw = 0.0797
Create an exhaustive searcher object e from the data K.
tice = ExhaustiveSearcher(K);ces = toc
ces = 0.0021
Creating an exhaustive searcher is much faster than creating an HNSW searcher.
Time the search using e and compare the result to the time using the HNSW searcher h.
tic;[idx0, D0] = knnsearch(e,Y,K=5);tes = toc
tes = 1.4841
In this case, the HNSW searcher computes about 20 times faster than the exhaustive searcher.
Determine how many results of the HNSW search differ in some way from the results of the exhaustive search.
v = abs(idx - idx0); % Count any difference in the five found entriesvv = sum(v,2); % vv = 0 means no differencenz = nnz(vv) % Out of 1000 rows how many differ at all?
nz = 118
Here, 118 of 1000 HNSW search results differ from the exhaustive search results.
Try to improve the accuracy of the HNSW searcher by training with nondefault parameters. Specifically, use larger values for MaxNumLinksPerNode and TrainSetSize. These settings affect the speed of training and finding nearest neighbors.
tic;h2 = hnswSearcher(K,MaxNumLinksPerNode=48,TrainSetSize=2000);chnsw2 = toc
chnsw2 = 78.4836
With these parameters, the searcher takes about seven times as long to train.
tic;[idx2, D2] = knnsearch(h2,Y,K=5);thnsw2 = toc
thnsw2 = 0.1049
The speed of finding nearest neighbors using HNSW decreases, but is still more than ten times faster than the exhaustive search.
v = abs(idx2 - idx0);vv = sum(v,2);nz = nnz(vv)
nz = 57
For the slower but more accurate HNSW search, only 57 of 1000 results differ in any way from the exact results. Summarize the timing results in a table.
timet = table([chnsw;ces;chnsw2],[thnsw;tes;thnsw2],'RowNames',["HNSW";"Exhaustive";"HNSW2"],'VariableNames',["Creation" "Search"])
timet=3×2 table Creation Search _________ ________ HNSW 10.654 0.079741 Exhaustive 0.0021304 1.4841 HNSW2 78.484 0.10492Input Arguments
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Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'K',2,'Distance','minkowski' specifies to find the two nearest neighbors of Mdl.X to each point in Y and to use the Minkowski distance metric.
For Exhaustive Nearest Neighbor Searchers and Kd-Tree Nearest Neighbor Searchers
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IncludeTies — Flag to include all nearest neighbors
false (0) (default) | true (1)
Flag to include nearest neighbors that have the same distance from query observations, specified as the comma-separated pair consisting of 'IncludeTies' and false (0) or true (1).
If IncludeTies is true, then:
knnsearchincludes all nearest neighbors whose distances are equal to the kth smallest distance in the output arguments, where k is the number of searched nearest neighbors specified by the 'K' name-value pair argument.Idx and D are m-by-
1cell arrays such that each cell contains a vector of at least k indices and distances, respectively. Each vector inDcontains arranged distances in ascending order. Each row inIdxcontains the indices of the nearest neighbors corresponding to the distances inD.
If IncludeTies is false, then knnsearch chooses the observation with the smallest index among the observations that have the same distance from a query point.
Example: 'IncludeTies',true
K — Number of nearest neighbors
1 (default) | positive integer
Number of nearest neighbors to search for in the training data per query observation, specified as the comma-separated pair consisting of 'K' and a positive integer.
Example: 'K',2
Data Types: single | double
For Kd-Tree Nearest Neighbor Searchers
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SortIndices — Flag to sort returned indices according to distance
true (1) (default) | false (0)
Flag to sort returned indices according to distance, specified as the comma-separated pair consisting of 'SortIndices' and either true (1) or false (0).
For faster performance, you can set SortIndices to false when the following are true:
Y contains many observations that have many nearest neighbors in
X.Mdl is a
KDTreeSearchermodel object.IncludeTies is
false.
In this case, knnsearch returns the indices of the nearest neighbors in no particular order. When SortIndices is true, the function arranges the nearest neighbor indices in ascending order by distance.
SortIndices is true by default. When Mdl is an ExhaustiveSearcher model object or IncludeTies is true, the function always sorts the indices.
Example: 'SortIndices',false
Data Types: logical
For Exhaustive Nearest Neighbor Searchers
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For HNSW Searchers
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SearchSetSize — Size of candidate list of nearest neighbors
max(10,K) (default) | positive integer from K through N
Size of the candidate list of nearest neighbors for a single query point during the search process, specified as a positive integer from K through N. Larger values lead to a more complete search, which increases the likelihood of finding the true nearest neighbors, but slow the search. SearchSetSize must be at least K, the number of features in the data (number of columns in Mdl.X), and must be no more than N, the number of rows in the training data (number of rows in Mdl.X).
Example: SearchSetSize=15
Data Types: double
Note
If you specify 'Distance', 'Cov', 'P', or 'Scale', then Mdl.Distance and Mdl.DistParameter do not change value.
Output Arguments
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Idx — Training data indices of nearest neighbors
numeric matrix | cell array of numeric vectors
Training data indices of nearest neighbors, returned as a numeric matrix or cell array of numeric vectors.
If you do not specify IncludeTies (
falseby default), thenIdxis an m-by-k numeric matrix, where m is the number of rows in Y and k is the number of searched nearest neighbors specified by the 'K' name-value pair argument.Idx(j,i)indicates thatMdl.X(Idx(j,i),:)is one of the k closest observations inMdl.Xto the query observationY(j,:).If you specify
'IncludeTies',true, thenIdxis an m-by-1cell array such that cellj(Idx{j}) contains a vector of at least k indices of the closest observations inMdl.Xto the query observationY(j,:).
If SortIndices is true, then knnsearch arranges the indices in ascending order by distance.
D — Distances of nearest neighbors
numeric matrix | cell array of numeric vectors
Distances of the nearest neighbors to the query data, returned as a numeric matrix or cell array of numeric vectors.
If you do not specify IncludeTies (
falseby default), thenDis an m-by-k numeric matrix, where m is the number of rows in Y and k is the number of searched nearest neighbors specified by the 'K' name-value pair argument.D(j,i)is the distance betweenMdl.X(Idx(j,i),:)and the query observationY(j,:)with respect to the distance metric.If you specify
'IncludeTies',true, thenDis an m-by-1cell array such that cellj(D{j}) contains a vector of at least k distances of the closest observations inMdl.Xto the query observationY(j,:).
If SortIndices is true, then knnsearch arranges the distances in ascending order.
Tips
knnsearch finds the k (positive integer) points in Mdl.X that are k-nearest for each Y point. In contrast, rangesearch finds all the points in Mdl.X that are within distance r (positive scalar) of each Y point.
Algorithms
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Fast Euclidean Distance Algorithm
The values of the Distance argument that begin fast (such as 'fasteuclidean' and 'fastseuclidean') calculate Euclidean distances using an algorithm that uses extra memory to save computational time. This algorithm is named "Euclidean Distance Matrix Trick" in Albanie [1] and elsewhere. Internal testing shows that this algorithm saves time when the number of predictors is at least 10. Algorithms starting with 'fast' do not support sparse data.
To find the matrix D of distances between all the points xi and xj, where each xi has n variables, the algorithm computes distance using the final line in the following equations:
The matrix in the last line of the equations is called the Gram matrix. Computing the set of squared distances is faster, but slightly less numerically stable, when you compute and use the Gram matrix instead of computing the squared distances by squaring and summing. For a discussion, see Albanie [1].
References
[1] Albanie, Samuel. Euclidean Distance Matrix Trick. June, 2019. Available at https://www.robots.ox.ac.uk/%7Ealbanie/notes/Euclidean_distance_trick.pdf.
Alternative Functionality
knnsearchis an object function that requires an ExhaustiveSearcher, KDTreeSearcher, or hnswSearcher model object and query data. Under equivalent conditions for an exhaustive or Kd-tree search, theknnsearchobject function returns the same results as the knnsearch function with the specified name-value argumentNSMethod="exhaustive"orNSMethod="kdtree", respectively. Note that theknnsearchfunction does not provide a similar name-value argument for specifying anhnswSearchermodel.For k-nearest neighbors classification, see fitcknn and ClassificationKNN.
References
[1] Friedman, J. H., Bentley, J., and Finkel, R. A. (1977). “An Algorithm for Finding Best Matches in Logarithmic Expected Time.” ACM Transactions on Mathematical Software Vol. 3, Issue 3, Sept. 1977, pp. 209–226.
[2] Malkov, Yu. A., and D. A. Yashunin. Efficient and Robust Approximate Nearest Neighbor Search Using Hierarchical Navigable Small World Graphs. Available at https://arxiv.org/abs/1603.09320.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
This table contains notes about the arguments of
knnsearch. Arguments not included in this table are fully supported.Argument Notes and Limitations MdlIf
Mdlis an hnswSearcher object, you cannot useMdlfor code generation.There are two ways to use
Mdlin code generation. For an example, see Code Generation for Nearest Neighbor Searcher.Use saveLearnerForCoder, loadLearnerForCoder, and codegen (MATLAB Coder) to generate code for the
knnsearchfunction. Save a trained model by usingsaveLearnerForCoder. Define an entry-point function that loads the saved model by usingloadLearnerForCoderand calls theknnsearchfunction. Then usecodegento generate code for the entry-point function.Include
coder.Constant(Mdl)in the-argsvalue of codegen (MATLAB Coder).
If
Mdlis a KDTreeSearcher object, and the code generation build type is a MEX function, then codegen (MATLAB Coder) generates a MEX function using Intel® Threading Building Blocks (TBB) for parallel computation. Otherwise,codegengenerates code using parfor (MATLAB Coder).MEX function for the kd-tree search algorithm —
codegengenerates an optimized MEX function using Intel TBB for parallel computation on multicore platforms. You can use the MEX function to accelerate MATLAB® algorithms. For details on Intel TBB, see https://www.intel.com/content/www/us/en/developer/tools/oneapi/onetbb.html.If you generate the MEX function to test the generated code of the
parforversion, you can disable the usage of Intel TBB. Set theExtrinsicCallsproperty of the MEX configuration object tofalse. For details, see coder.MexCodeConfig (MATLAB Coder).MEX function for the exhaustive search algorithm and standalone C/C++ code for both algorithms — The generated code of
knnsearchuses parfor (MATLAB Coder) to create loops that run in parallel on supported shared-memory multicore platforms in the generated code. If your compiler does not support the Open Multiprocessing (OpenMP) application interface or you disable OpenMP library, MATLAB Coder™ treats theparfor-loops asfor-loops. To find supported compilers, see Supported Compilers. To disable OpenMP library, set theEnableOpenMPproperty of the configuration object tofalse. For details, see coder.CodeConfig (MATLAB Coder).
'Distance'Cannot be a custom distance function.
Must be a compile-time constant; its value cannot change in the generated code.
knnsearchdoes not support code generation for fast Euclidean distance computations, meaning those distance metrics whose names begin withfast(for example,'fasteuclidean').
'IncludeTies'Must be a compile-time constant; its value cannot change in the generated code.
'SortIndices'Not supported. The output arguments are always sorted. Name-value pair arguments Names in name-value arguments must be compile-time constants. For example, to allow a user-defined exponent for the Minkowski distance in the generated code, include
{coder.Constant('Distance'),coder.Constant('Minkowski'),coder.Constant('P'),0}in the-argsvalue of codegen (MATLAB Coder).IdxWhen you specify
'IncludeTies'astrue, the sorted order of tied distances in the generated code can be different from the order in MATLAB due to numerical precision.knnsearchreturns integer-type (int32) indices in generated standalone C/C++ code. Therefore, the function allows for strict single-precision support when you use single-precision inputs. For MEX code generation, the function still returns double-precision indices to match the MATLAB behavior.Before R2020a:
knnsearchreturns double-precision indices in generated standalone C/C++ code.
For more information, see Introduction to Code Generation and Code Generation for Nearest Neighbor Searcher.
Version History
Introduced in R2010a
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R2024a: Approximate search using hnswSearcher object
Find approximate nearest neighbors using an hnswSearcher model object. An hnswSearcher object takes longer to create than an ExhaustiveSearcher or KDTreeSearcher object, but computes approximate nearest neighbors more quickly. Create an hnswSearcher object by using the hnswSearcher function or by using createns with the specification NSMethod="hnsw". For details, see hnswSearcher.
R2023a: Fast Euclidean distance using a cache
The 'fasteuclidean' and 'fastseuclidean' distance metrics accelerate the computation of Euclidean distances by using a cache and a different algorithm (see Algorithms).
See Also
createns | ExhaustiveSearcher | KDTreeSearcher | hnswSearcher | rangesearch | knnsearch | fitcknn | ClassificationKNN
Topics
- k-Nearest Neighbor Search and Radius Search
- Distance Metrics
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