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Nearest neighbor classification CSE 250B
创建日期：2024-04-02 15:50:16
所属分类：物理Physical
标签： CSE 250B
Nearest neighbor classification
Nearest neighbor classification
CSE 250B
The problem we’ll solve today
Given an image of a handwritten digit, say which digit it is.
=⇒ 3
Some more examples:
The problem we’ll solve today
Given an image of a handwritten digit, say which digit it is.
=⇒ 3
Some more examples:
The machine learning approach
Assemble a data set:
The MNIST data set of handwritten digits:
• Training set of 60,000 images and their labels.
• Test set of 10,000 images and their labels.
And let the machine figure out the underlying patterns.
Nearest neighbor classification
Training images x (1), x (2), x (3), . . . , x (60000)
Labels y (1), y (2), y (3), . . . , y (60000) are numbers in the range 0− 9
How to classify a new image x?
• Find its nearest neighbor amongst the x (i)
• Return y (i)
The data space
How to measure the distance between images?
MNIST images:
• Size 28× 28 (total: 784 pixels)
• Each pixel is grayscale: 0-255
Stretch each image into a vector with 784 coordinates:
• Data space X = R784
• Label space Y = {0, 1, . . . , 9}
The data space
How to measure the distance between images?
MNIST images:
• Size 28× 28 (total: 784 pixels)
• Each pixel is grayscale: 0-255
Stretch each image into a vector with 784 coordinates:
• Data space X = R784
• Label space Y = {0, 1, . . . , 9}
The distance function
Remember Euclidean distance in two dimensions?
x = (1, 2)
z = (3, 5)
Euclidean distance in higher dimension
Euclidean distance between 784-dimensional vectors x , z is
‖x − z‖ =
√√√√ 784∑
i=1
(xi − zi )2
Here xi is the ith coordinate of x .
Nearest neighbor classification
Training images x (1), . . . , x (60000), labels y (1), . . . , y (60000)
To classify a new image x :
• Find its nearest neighbor amongst the x (i) using
Euclidean distance in R784
• Return y (i)
How accurate is this classifier?
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points?
Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?) 90%.
• Test error of nearest neighbor: 3.09%.
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points? Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?) 90%.
• Test error of nearest neighbor: 3.09%.
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points? Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?) 90%.
• Test error of nearest neighbor: 3.09%.
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points? Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?)
90%.
• Test error of nearest neighbor: 3.09%.
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points? Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?) 90%.
• Test error of nearest neighbor: 3.09%.
Accuracy of nearest neighbor on MNIST
Training set of 60,000 points.
• What is the error rate on training points? Zero.
In general, training error is an overly optimistic predictor of future
performance.
• A better gauge: separate test set of 10,000 points.
Test error = fraction of test points incorrectly classified.
• What test error would we expect for a random classifier?
(One that picks a label 0− 9 at random?) 90%.
• Test error of nearest neighbor: 3.09%.
Examples of errors
Test set of 10,000 points:
• 309 are misclassified
• Error rate 3.09%
Examples of errors:
Query
NN
Ideas for improvement: (1) k-NN (2) better distance function.
Examples of errors
Test set of 10,000 points:
• 309 are misclassified
• Error rate 3.09%
Examples of errors:
Query
NN
Ideas for improvement: (1) k-NN (2) better distance function.
K -nearest neighbor classification
Classify a point using the labels of its k nearest neighbors among the
training points.
MNIST:
k 1 3 5 7 9 11
Test error (%) 3.09 2.94 3.13 3.10 3.43 3.34
In real life, there’s no test set. How to decide which k is best?
1 Hold-out set.
• Let S be the training set.
• Choose a subset V ⊂ S as a validation set.
• What fraction of V is misclassified by finding the k-nearest
neighbors in S \ V ?
2 Leave-one-out cross-validation.
• For each point x ∈ S , find the k-nearest neighbors in S \ {x}.
• What fraction are misclassified?
K -nearest neighbor classification
Classify a point using the labels of its k nearest neighbors among the
training points.
MNIST:
k 1 3 5 7 9 11
Test error (%) 3.09 2.94 3.13 3.10 3.43 3.34
In real life, there’s no test set. How to decide which k is best?
1 Hold-out set.
• Let S be the training set.
• Choose a subset V ⊂ S as a validation set.
• What fraction of V is misclassified by finding the k-nearest
neighbors in S \ V ?
2 Leave-one-out cross-validation.
• For each point x ∈ S , find the k-nearest neighbors in S \ {x}.
• What fraction are misclassified?
K -nearest neighbor classification
Classify a point using the labels of its k nearest neighbors among the
training points.
MNIST:
k 1 3 5 7 9 11
Test error (%) 3.09 2.94 3.13 3.10 3.43 3.34
In real life, there’s no test set. How to decide which k is best?
1 Hold-out set.
• Let S be the training set.
• Choose a subset V ⊂ S as a validation set.
• What fraction of V is misclassified by finding the k-nearest
neighbors in S \ V ?
2 Leave-one-out cross-validation.
• For each point x ∈ S , find the k-nearest neighbors in S \ {x}.
• What fraction are misclassified?
K -nearest neighbor classification
Classify a point using the labels of its k nearest neighbors among the
training points.
MNIST:
k 1 3 5 7 9 11
Test error (%) 3.09 2.94 3.13 3.10 3.43 3.34
In real life, there’s no test set. How to decide which k is best?
1 Hold-out set.
• Let S be the training set.
• Choose a subset V ⊂ S as a validation set.
• What fraction of V is misclassified by finding the k-nearest
neighbors in S \ V ?
2 Leave-one-out cross-validation.
• For each point x ∈ S , find the k-nearest neighbors in S \ {x}.
• What fraction are misclassified?
K -nearest neighbor classification
Classify a point using the labels of its k nearest neighbors among the
training points.
MNIST:
k 1 3 5 7 9 11
Test error (%) 3.09 2.94 3.13 3.10 3.43 3.34
In real life, there’s no test set. How to decide which k is best?
1 Hold-out set.
• Let S be the training set.
• Choose a subset V ⊂ S as a validation set.
• What fraction of V is misclassified by finding the k-nearest
neighbors in S \ V ?
2 Leave-one-out cross-validation.
• For each point x ∈ S , find the k-nearest neighbors in S \ {x}.
• What fraction are misclassified?
Cross-validation
How to estimate the error of k-NN for a particular k?
10-fold cross-validation
• Divide the training set into 10 equal pieces.
Training set (call it S): 60,000 points
Call the pieces S1,S2, . . . ,S10: 6,000 points each.
• For each piece Si :
• Classify each point in Si using k-NN with training set S − Si
• Let i = fraction of Si that is incorrectly classified
• Take the average of these 10 numbers:
estimated error with k-NN =
1 + · · ·+ 10
10
Another improvement: better distance functions
The Euclidean (`2) distance between these two images is very high!
Much better idea: distance measures that are invariant under:
• Small translations and rotations. e.g. tangent distance.
• A broader family of natural deformations. e.g. shape context.
Test error rates:
`2 tangent distance shape context
3.09 1.10 0.63
Another improvement: better distance functions
The Euclidean (`2) distance between these two images is very high!
Much better idea: distance measures that are invariant under:
• Small translations and rotations. e.g. tangent distance.
• A broader family of natural deformations. e.g. shape context.
Test error rates:
`2 tangent distance shape context
3.09 1.10 0.63
Another improvement: better distance functions
The Euclidean (`2) distance between these two images is very high!
Much better idea: distance measures that are invariant under:
• Small translations and rotations. e.g. tangent distance.
• A broader family of natural deformations. e.g. shape context.
Test error rates:
`2 tangent distance shape context
3.09 1.10 0.63
Related problem: feature selection
Feature selection/reweighting is part of picking a distance function.
And, one noisy feature can wreak havoc with nearest neighbor!
versus
Related problem: feature selection
Feature selection/reweighting is part of picking a distance function.
And, one noisy feature can wreak havoc with nearest neighbor!
versus
Algorithmic issue: speeding up NN search
Naive search takes time O(n) for training set of size n: slow!
There are data structures for speeding up nearest neighbor search, like:
1 Locality sensitive hashing
2 Ball trees
3 K -d trees
These are part of standard Python libraries for NN, and help a lot.
Algorithmic issue: speeding up NN search
Naive search takes time O(n) for training set of size n: slow!
There are data structures for speeding up nearest neighbor search, like:
1 Locality sensitive hashing
2 Ball trees
3 K -d trees
These are part of standard Python libraries for NN, and help a lot.
Example: k-d trees for NN search
A hierarchical, rectilinear spatial partition.
For data set S ⊂ Rd :
• Pick a coordinate 1 ≤ i ≤ d .
• Compute v = median({xi : x ∈ S}).
• Split S into two halves:
SL = {x ∈ S : xi < v}
SR = {x ∈ S : xi ≥ v}

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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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