Original: Generalisation and the Geometry of Class Separability
Revised: Generalisation and the Geometry of Class Separability

Original: Abstract
Revised: Abstract

Original: Recent results in deep learning show that considering only the capacity of machinesdoes not adequately explain the generalisation performance we can observe [7, 11].
Revised: Recent results in deep learning show that considering only the capacity of machines does not adequately explain the generalisation performance we can observe [7, 11].

Original: We propose that by considering the geometry of the data we can better explaingeneralisation achieved in deep learning.
Revised: We propose that by considering the geometry of the data we can better explain generalisation achieved in deep learning.

Original: In particular we show that in classiﬁcationthe separability of the data can explain how good generalisation can be achieved inhigh dimensions.
Revised: In particular we show that in classiﬁcation the separability of the data can explain how good generalisation can be achieved in high dimensions.

Original: Further we show that layers within a CNNs sequentially increasethe linear separability of data, and that the information these layers retain or discardcan help explain why these models generalise.
Revised: Further we show that layers within a CNNs sequentially increase the linear separability of data, and that the information these layers retain or discard can help explain why these models generalise.

Original:  Introduction
Revised:  Introduction

Original: Existing work studying generalisation in machine learning has typically focused on the properties oflearning machines, most signiﬁcantly on their capacity and the size of the hypothesis class [4, 6, 9].
Revised: Existing work studying generalisation in machine learning has typically focused on the properties of learning machines, most signiﬁcantly on their capacity and the size of the hypothesis class [4, 6, 9].

Original: Relatively little work has explored the geometry of the data, and how this impacts the generalisationperformance in different settings.
Revised: Relatively little work has explored the geometry of the data, and how this impacts the generalisation performance in different settings.

Original: Empirical results show that the capacity of a machine is not by itselfa good indicator of generalisation performance [7], and therefore focusing only on the hypothesisclass does not give the full picture of generalisation in a given setting.
Revised: Empirical results show that the capacity of a machine is not by itself a good indicator of generalisation performance [7], and therefore focusing only on the hypothesis class does not give the full picture of generalisation in a given setting.

Original: It is therefore important toinvestigate how the data impacts generalisation.
Revised: It is therefore important to investigate how the data impacts generalisation.

Original: In many cases deep learning is applied to data which live in a large number of dimensions [2], inwhich it is difﬁcult to make sense of its geometry.
Revised: In many cases deep learning is applied to data which live in a large number of dimensions [2], in which it is difﬁcult to make sense of its geometry.

Original: A key observation is that the distance between datapoints has little meaning in high dimensions, since this grows with the square root of the number ofdimensions, making similar data points appear distant [1].
Revised: A key observation is that the distance between data points has little meaning in high dimensions, since this grows with the square root of the number of dimensions, making similar data points appear distant [1].

Original: We hypothesise that in many different settings our real data lives on a low dimensional manifoldwithin the data space, and that the geometry of this manifold is directly relevant to both the suitabilityof different machines and the generalisation performance these can obtain.
Revised: We hypothesise that in many different settings our real data lives on a low dimensional manifold within the data space, and that the geometry of this manifold is directly relevant to both the suitability of different machines and the generalisation performance these can obtain.

Original: Moreover we believethat how the geometry of this manifold is transformed by a machine is key to understanding theinformation captured by the machine, and therefore also to understanding how certain machines areable to generalise.
Revised: Moreover we believe that how the geometry of this manifold is transformed by a machine is key to understanding the information captured by the machine, and therefore also to understanding how certain machines are able to generalise.

Original: This is of particular signiﬁcance in the case of Deep Learning, in which data issuccessively transformed by many functions to yield the output [2].
Revised: This is of particular signiﬁcance in the case of Deep Learning, in which data is successively transformed by many functions to yield the output [2].

Original:  Our work
Revised:  Our work

Original: We investigate projecting the data into a low dimensional subspace in order to give insight into itsgeometry, in particular in classiﬁcation, where we aim to ﬁnd a projection which captures the speciﬁcgeometry of class separation.
Revised: We investigate projecting the data into a low dimensional subspace in order to give insight into its geometry, in particular in classiﬁcation, where we aim to ﬁnd a projection which captures the speciﬁc geometry of class separation.

Original: We also present the idea that in the case of classiﬁcation the separabilityof the data has an important role in generalisation, and show that the separation of classes can yieldgood generalisation even in high dimensions.
Revised: We also present the idea that in the case of classiﬁcation the separability of the data has an important role in generalisation, and show that the separation of classes can yield good generalisation even in high dimensions.

Original: 2.1 Theoretical results on class separation
Revised: 2.1 Theoretical results on class separation

Original: We consider the case where we have data consisting of p -dimensional feature vectors X and the taskis to classify them into two classes ± 1 .
Revised: We consider the case where we have data consisting of p -dimensional feature vectors X and the task is to classify them into two classes ± 1 .

Original: We assume the data points, ( X , Y ) , are distributed accordingto p X ,Y ( x , y )
Revised: We assume the data points, ( X , Y ) , are distributed according to p X ,Y ( x , y )

Original: =
Revised: =

Original: p X | Y ( x | y) P Y
Revised: p X | Y ( x | y

Original: [ Y =y ]
Revised: [Equation] Y =

Original: [Equation]
Revised: [Equation]

Original: where (cid:74) predicate (cid:75) is an indicator function, ∆ is the distance along some x ∗ ∈ R p from the mean ofeach class to the origin, such that the means are separated by 2∆ , and | x ∗ | = 1 .
Revised: where (cid:74) predicate (cid:75) is an indicator function, ∆ is the distance along some x ∗ ∈ R p from the mean of each class to the origin, such that the means are separated by 2∆ , and | x ∗ | = 1 .

Original: We assume that wehave a training data set, D , consisting of m examples drawn independently from p X ,Y ( x , y ) .
Revised: We assume that we have a training data set, D , consisting of m examples drawn independently from p X ,Y ( x , y ) .

Original: We consider a perceptron with weight vectors w h ∈ S p (the p -sphere) that makes predictionsh ( X ) = sign( w T h X ) .
Revised: We consider a perceptron with weight vectors w h ∈ S p (the p -sphere) that makes predictions h ( X ) = sign( w T h X ) .

Original: We deﬁne a loss function for a hypothesis h (i.e. the perceptron with weightsw h ) to be
Revised: We deﬁne a loss function for a hypothesis h (i.e. the perceptron with weights w h ) to be

Original: [Equation]
Revised: [Equation]

Original: The risk of a hypothesis, h , is then given by
Revised: The risk of a hypothesis, h , is then given by

Original: [Equation]
Revised: [Equation]

Original:  where cos( θ h ) =
Revised: (2) where cos( θ h ) =

Original: w T h x ∗ and Φ( z ) is the cumulative distribution function for a standard normaldistribution..
Revised: w T h x ∗ and Φ( z ) is the cumulative distribution function for a standard normal distribution..

Original: We assume are are given m training examples { ( X α , Y α ) | α = 1 , 2 , . . .
Revised: We assume are are given m training examples { ( X α , Y α ) | α = 1 , 2 , . . .

Original: , m } drawn independentlyfrom p X ,Y .
Revised: , m } drawn independently from p X ,Y .

Original: We consider a Hebb classiﬁer where we choose a weight vector w = ¯ w / | ¯ w | where¯
Revised: We consider a Hebb classiﬁer where we choose a weight vector w = ¯ w / | ¯ w | where

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Given that the risk of a weight vector with an angle of θ from x ∗ is Φ( − ∆ cos( θ )) then the expectedrisk using the Hebb rule can be approximated as
Revised: Given that the risk of a weight vector with an angle of θ from x ∗ is Φ( − ∆ cos( θ )) then the expected risk using the Hebb rule can be approximated as

Original: [Equation]
Revised: [Equation]

Original: In Figure 1 we show the expected risk versus the number of training examples for feature vectorsof length p = 50 , 100 and 200 and for different separations of the two distribution ∆ = 1 , 2 , 3 and4.
Revised: In Figure 1 we show the expected risk versus the number of training examples for feature vectors of length p = 50 , 100 and 200 and for different separations of the two distribution ∆ = 1 , 2 , 3 and 4.

Original: In addition, we show simulation results averaged over 100 different training runs.
Revised: In addition, we show simulation results averaged over 100 different training runs.

Original: Clearly, thisapproximation provides a very good approximation to the observed behaviour.
Revised: Clearly, this approximation provides a very good approximation to the observed behaviour.

Original: 2.2 Empirical work on CNNs
Revised: 2.2 Empirical work on CNNs

Original: Our work focuses on the application of CNNs in image classiﬁcation.
Revised: Our work focuses on the application of CNNs in image classiﬁcation.

Original: Image data exempliﬁes the lowdimensional geometry of the true data which we are interested in, as is evidenced by the fact thatrandomly sampling the image space would never produce anything resembling a recognisable image.
Revised: Image data exempliﬁes the low dimensional geometry of the true data which we are interested in, as is evidenced by the fact that randomly sampling the image space would never produce anything resembling a recognisable image.

Original: We can also show by means of a simple example that classes within image data are not linearlyseparable, as is shown in the separability of the raw data in ﬁgure 2.
Revised: We can also show by means of a simple example that classes within image data are not linearly separable, as is shown in the separability of the raw data in ﬁgure 4.

Original: CNNs are of speciﬁc interest as they seem to be particularly well attuned for use on image data, asis clear from the multitude of recent work showcasing exceptional performance [5, 3, 10].

Original: Indeedthe state of the art performance on ImageNet is achieved by a CNN with 480 million parameters[8], exemplifying the point that capacity is not a good indicator for generalisation.

Original: We hypothesisethat the geometry of real image data is directly related to the suitability of CNNs to images, and webelieve that the way in which convolutional layers transform the data is of particular signiﬁcance inunderstanding how CNNs achieve such exceptional generalisation performance.

Original: As we have shown the signiﬁcance of separability in generalisation, we investigate the linear separability of data as it is transformed by a CNN.

Original: In order to achieve good classiﬁcation performance,the data must become linearly separable in the ﬁnal layer, however there is no requirement for linearseparability at any previous layer, due to the use of nonlinear activations.

Original: Nevertheless, increasing

Original: [Figure]
Revised: [Figure]

Original: ∆ =
Revised: ∆ =

Original: ∆
Revised: ∆

Original: [Figure]
Revised: [Figure]

Original: ∆
Revised: ∆

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: m
Revised: m

Original: [Figure]
Revised: [Figure]

Revised: CNNs are of speciﬁc interest as they seem to be particularly well attuned for use on image data, as is clear from the multitude of recent work showcasing exceptional performance [5, 3, 10].

Revised: Indeed the state of the art performance on ImageNet is achieved by a CNN with 480 million parameters [8], exemplifying the point that capacity is not a good indicator for generalisation.

Revised: We hypothesise that the geometry of real image data is directly related to the suitability of CNNs to images, and we believe that the way in which convolutional layers transform the data is of particular signiﬁcance in understanding how CNNs achieve such exceptional generalisation performance.

Revised: As we have shown the signiﬁcance of separability in generalisation, we investigate the linear separability of data as it is transformed by a CNN.

Revised: In order to achieve good classiﬁcation performance, the data must become linearly separable in the ﬁnal layer, however there is no requirement for linear separability at any previous layer, due to the use of nonlinear activations.

Original:  linear separability would indicate that the convolutional layers transform the geometry of the data ina way that enables good generalisation.
Revised: Nevertheless, increasing linear separability would indicate that the convolutional layers transform the geometry of the data in a way that enables good generalisation.

Original: We take a trained CNN F : X → Y , and split it at some layer l , to give us a network F l : X → Z l ,with F l being the network given by the layers of F up to and including l .
Revised: We take a trained CNN F : X → Y , and split it at some layer l , to give us a network F l : X → Z l , with F l being the network given by the layers of F up to and including l .

Original: We call Z l the Representation
Revised: We call Z l the Representation

Original: Space of F l .
Revised: Space of F l .

Original: We then map the data using F l to give us a re-representation of this data in the
Revised: We then map the data using F l to give us a re-representation of this data in the

Original: Representation Space.
Revised: Representation Space.

Original: By projecting the data in each Representation Space into a low dimensionalsubspace, we can examine how salient information is extracted throughout the network, and spuriousinformation is ﬁltered out.
Revised: By projecting the data in each Representation Space into a low dimensional subspace, we can examine how salient information is extracted throughout the network, and spurious information is ﬁltered out.

Original: We then train a linear classiﬁer on the re-represented data in the Representation Space, and take thetest accuracy of the classiﬁer to be an estimate of the relative linear separability of the data.
Revised: We then train a linear classiﬁer on the re-represented data in the Representation Space, and take the test accuracy of the classiﬁer to be an estimate of the relative linear separability of the data.

Original: In a
Revised: In a

Original: C-class problem a linear classiﬁer is simply a projection into C dimensions, which aims to capture asmuch salient information as possible.
Revised: C-class problem a linear classiﬁer is simply a projection into C dimensions, which aims to capture as much salient information as possible.

Original: We experiment using CNNs of varying depths trained on CIFAR10.
Revised: We experiment using CNNs of varying depths trained on CIFAR10.

Original: We term an N layer CNN tobe a CNN with N convolutional layers.
Revised: We term an N layer CNN to be a CNN with N convolutional layers.

Original: For the purposes of splitting the network we consider eachoperation, such as a convolution, activation, or max pooling, to be its own layer, as we are interestedin how each of these transform the data.
Revised: For the purposes of splitting the network we consider each operation, such as a convolution, activation, or max pooling, to be its own layer, as we are interested in how each of these transform the data.

Original:  Results
Revised:  Results

Original: We look at three CNN models, of 4, 6 and 9 convolutional layers.
Revised: We look at three CNN models, of 4, 6 and 9 convolutional layers.

Original: Figure 2 shows plots of theseparability on the training and test data, directly after each layer in each of these models.
Revised: Figure 2 shows the percentage change in separability in the 9 layer model between each layer, measured on the training and test data. Each layer corresponds to the change between that layer and the one immediately preceeding it.

Original: Theseresults are averaged over 15 trials.
Revised: These results are averaged over 15 trials.

Revised: See appendix A for results from all models.

Original: Our results demonstrate a clear trend of the linear separability increasing throughout the network, bothon the training and test data, giving a clear indication that the geometry of the data is becoming moreseparated.
Revised: Our results demonstrate a clear trend of the linear separability increasing throughout the network, both on the training and test data, giving a clear indication that the geometry of the data is becoming more separated.

Original: We observe that this increase is not monotonic, especially in the training data separability.
Revised: We observe that this increase is not monotonic, especially in the training data separability.

Original: There is a clear trend mirrored across each network of the separability decreasing with each convolution, followed by a sharp increase following the activation function.
Revised: There is a clear trend mirrored across each network of the separability decreasing with each convolution, followed by a sharp increase following the activation function.

Original: We see this trend in boththe training and test data, however it is much more pronounced in the training data, which may beindicative of overﬁtting, in that the network is extracting some spurious information on the trainingdata as well as salient information on the underlying distribution.
Revised: We see this trend in both the training and test data, however it is much more pronounced in the training data, which may be indicative of overﬁtting, in that the network is extracting some spurious information on the training data as well as salient information on the underlying distribution.

Original: It seems clear that the convolutionand activation work in tandem to augment the data in such a way as to make it more separable, therebygiving improved performance, but that this can overﬁt to spurious information within the trainingdata.
Revised: It seems clear that the convolution and activation work in tandem to augment the data in such a way as to make it more separable, thereby giving improved performance, but that this can overﬁt to spurious information within the training data.

Original: [Figure]

Original: In contrast, we also observe that the max pooling layers induce signiﬁcant decreases in the separabilityof the training data, but increases in separability of the test data.
Revised: In contrast, we also observe that the max pooling layers induce signiﬁcant decreases in the separability of the training data, but increases in separability of the test data.

Original: This suggests that the maxpooling layers help to prevent overﬁtting, by discarding spurious information but retaining salientinformation.
Revised: This suggests that the max pooling layers help to prevent overﬁtting, by discarding spurious information but retaining salient information.

Original: We can interpret max pooling layers as focusing on strong responses to ﬁlters, indicatingthe presence of particular features, but discarding ﬁne-grained positional information.
Revised: We can interpret max pooling layers as focusing on strong responses to ﬁlters, indicating the presence of particular features, but discarding ﬁne-grained positional information.

Original: This issomewhat unsurprising, as the position of an object is not relevant for classifying it.
Revised: This is somewhat unsurprising, as the position of an object is not relevant for classifying it.

Original: This suggeststhat max pooling plays an important role in transforming the geometry of the data to improvegeneralisation.
Revised: This suggests that max pooling plays an important role in transforming the geometry of the data to improve generalisation.

Original: Our results are a strong indication that linear separability emerges throughout the subsequent layersof a CNN, and we believe this to be an important factor in the exceptional generalisation performancethese models can achieve.
Revised: Our results are a strong indication that linear separability emerges throughout the subsequent layers of a CNN, and we believe this to be an important factor in the exceptional generalisation performance these models can achieve.

Original:  Conclusions
Revised:  Conclusions

Original: Our results show that the geometry of class separation in data is signiﬁcant in the generalisationperformance which can be obtained in learning.
Revised: Our results show that the geometry of class separation in data is signiﬁcant in the generalisation performance which can be obtained in learning.

Original: Moreover our results show how seperability emergesin sequential layers of a trained deep network, giving insight into why deep networks are able togeneralise despite over-parameterisation.
Revised: Moreover our results show how seperability emerges in sequential layers of a trained deep network, giving insight into why deep networks are able to generalise despite over-parameterisation.

Original: Speciﬁcally we have shown that particular operators in anetwork contribute to this in different ways, in particular that max pooling has an important propertyof distilling salient information from data, giving improved separability.
Revised: Speciﬁcally we have shown that particular operators in a network contribute to this in different ways, in particular that max pooling has an important property of distilling salient information from data, giving improved separability.

Original: We believe the hierarchicalconstruction of complex features exhibited by deep networks is directly related to the process oflayers sequentially ﬁltering spurious information from data, which prevents deep networks overﬁttingand gives good generalisation.
Revised: We believe the hierarchical construction of complex features exhibited by deep networks is directly related to the process of layers sequentially ﬁltering spurious information from data, which prevents deep networks overﬁtting and gives good generalisation.

Original: We believe linear separability in the representation of the data to becharacteristic of this, as our results show linear separability to be both important for generalisaiton,and evident in deep networks.
Revised: We believe linear separability in the representation of the data to be characteristic of this, as our results show linear separability to be both important for generalisaiton, and evident in deep networks.

Original: Broader Impact
Revised: Broader Impact

Original: We believe this to not be applicable to our work.
Revised: We believe this to not be applicable to our work.

Revised: Appendix A Full Results

Revised: [Figure]

Revised: [Figure]