Original: Model Stitching: Looking For Functional Similarity Between Representations
Revised: Model Stitching: Looking For Functional Similarity Between Representations

Original: Abstract
Revised: Abstract

Original: Model stitching (Lenc & Vedaldi 2015) is a compelling methodology to comparedifferent neural network representations, because it allows us to measure to whatdegree they may be interchanged.
Revised: Model stitching (Lenc & Vedaldi 2015) is a compelling methodology to compare different neural network representations, because it allows us to measure to what degree they may be interchanged.

Original: We expand on a previous work from Bansal,
Revised: We expand on a previous work from Bansal,

Original: Nakkiran & Barak which used model stitching to compare representations of thesame shapes learned by differently seeded and/or trained neural networks of thesame architecture.
Revised: Nakkiran & Barak which used model stitching to compare representations of the same shapes learned by differently seeded and/or trained neural networks of the same architecture.

Original: Our contribution enables us to compare the representationslearned by layers with different shapes from neural networks with different architectures.
Revised: Our contribution enables us to compare the representations learned by layers with different shapes from neural networks with different architectures.

Original: We subsequently reveal unexpected behavior of model stitching.
Revised: We subsequently reveal unexpected behavior of model stitching.

Original: Namely,we ﬁnd that stitching, based on convolutions, for small ResNets, can reach highaccuracy if those layers come later in the ﬁrst (sender) network than in the second(receiver), even if those layers are far apart .
Revised: Namely, we ﬁnd that stitching, based on convolutions, for small ResNets, can reach high accuracy if those layers come later in the ﬁrst (sender) network than in the second (receiver), even if those layers are far apart .

Original: This leads us to hypothesize that stitches are not in fact learning to match the representations expected by receiver layers, but instead ﬁnding different representationswhich nonetheless yield similar results.
Revised: This leads us to hypothesize that stitches are not in fact learning to match the representations expected by receiver layers, but instead ﬁnding different representations which nonetheless yield similar results.

Original: Thus, we believe that model stitching maynot necessarily always be an accurate measure of similarity.
Revised: Thus, we suggest that model stitching, naively implemented, may not necessarily always be an accurate measure of similarity.

Original:  Introduction
Revised:  Introduction

Original: The success of deep learning for visual recognition has been attributed to the ability of neural networksto learn good representations of their training data Rumelhart et al.
Revised: The success of deep learning for visual recognition has been attributed to the ability of neural networks to learn good representations of their training data Rumelhart et al.

Original: That is, intermediateoutputs (which we refer to as “representations”) of good neural networks are believed to encodemeaningful information about their inputs, which these neural networks use for classiﬁcation and/orother downstream machine learning tasks Goodfellow et al. 
Revised: That is, intermediate outputs (which we refer to as “representations”) of good neural networks are believed to encode meaningful information about their inputs, which these neural networks use for classiﬁcation and/or other downstream machine learning tasks Goodfellow et al. (2016).

Original: However, our understanding ofthese representations is somewhat limited.
Revised: However, our understanding of these representations is somewhat limited.

Original: Though deep learning interpretability research, particularlyfor computer vision has helped us to intuitively grasp what deep neural networks are learning, we donot know why good representations are learned, nor do we have a robust theory to characterize them.
Revised: Though deep learning interpretability research, particularly for computer vision, has helped us to intuitively grasp what deep neural networks are learning, we do not know why good representations are learned, nor do we have a robust theory to characterize them.

Original: For example, we do not know how to compare representations effectively.
Revised: For example, we do not know how to compare representations effectively.

Original: Our goal is to improve the existing toolbox to ﬁnd functional similarity between representations.
Revised: Our goal is to improve the existing toolbox to ﬁnd functional similarity between representations.

Original: Itis not obvious how to ﬁnd functional similarity.
Revised: It is not obvious how to ﬁnd functional similarity, nor is it obvious exactly what it precisely means even though we (the authors) have some intuition of it, so before we continue we provide a sufﬁcient deﬁnition for our purposes. For us, a representational similarity metric is good at measuring “functional similarity” if we can easily use it to tell whether two representations of one or two models are used by the model(s) for the same or similar purpose(s). We believe this is a useful lens because if two representations can be used for similar purposes then in some sense they encode similar information.

Original: Many papers Kornblith et al.
Revised: While we are interested in functional similarity, many papers Kornblith et al.

Original: Morcos et al.(2018)
Revised: Morcos et al.

Original: Ding et al.
Revised: Ding et al.

Original: (2021) look for measures of statistical or geometric similarity  because they canconﬁrm known edge cases (i.e. when representations are exactly the same).
Revised: (2021) look for measures of statistical or geometric similarity 1 because they can conﬁrm known edge cases, like a pair of identical representations or a representation and a vector of random noise.

Original: However, these measures are  not very informative in general, since  neural networks’ representations can vary greatly in wayswhich are not yet well-understood, simply due to random initialization of the weights.
Revised: These measures are a great starting point, but are not very informative in general, since different neural networks could process and store information in ways which are analogous to eachother but not numerically similar.

Original: We believe that a previous work Bansal et al.
Revised: We believe that a previous work Bansal et al.

Original: (2021) provides us with such a measure.
Revised: (2021) provides us with such a measure.

Original: It uses,learned transformations to translate representations from one layer into those for another layer.
Revised: It uses, learned transformations to translate representations from one layer into those for another layer.

Original: Their technique measures functional similarity, because invariant to the type of transformations used, ittests whether two representations can be interchanged, which is a strong indicator that the tworepresentations function similarly.
Revised: Their technique measures functional similarity, because invariant to the type of transformations used, it tests whether two representations can be interchanged, which is a strong indicator that the two representations function similarly.

Original: However, their work can only compare representations with thesame shapes.
Revised: However, their work can only compare representations with the same shapes.

Original: We expand it to include all representations taking the form of ResNet tensors withwidths and heights that are multiples of each other.
Revised: We expand it to include all representations taking the form of ResNet tensors with widths and heights that are multiples of each other.

Original:  Experimental Setup2.1 Models and Dataset
Revised:  Experimental Setup 2.1 Models and Dataset

Original: We compare all different layers of all ResNets with a number of layers ranging from ten to eighteen.
Revised: We compare all different layers of all ResNets with a number of layers ranging from ten to eighteen.

Original: These ResNets are trained on CIFAR-10 for comparable results with Bansal et.
Revised: These ResNets are trained on CIFAR-10 for comparable results with Bansal et.

Original: al.
Revised: al.

Original: These small
Revised: These small

Original: ResNets we characterize with 4-tuples, where each element is either one or two, representing thenumber of residual blocks per stage 1 .
Revised: ResNets we characterize with 4-tuples, where each element is either one or two, representing the number of residual blocks per stage 2 .

Original: Since we use at most two blocks per stage, we can denote these4-tuples unambiguously as R 1111 , R 1112 , and so on.
Revised: Since we use at most two blocks per stage, we can denote these 4-tuples unambiguously as R 1111 , R 1112 , and so on.

Original: There are 2 4 = 16 such ResNets.
Revised: There are 2 4 = 16 such ResNets.

Original: Note that
Revised: Note that

Original: R 2222 is equivalent to the well-known Resnet18 architecture, while R 1111 is equivalent to Resnet10.
Revised: R 2222 is equivalent to the well-known Resnet18 architecture, while R 1111 is equivalent to Resnet10.

Original: 2.2 Experiment
Revised: 2.2 Experiment

Original: We train each possible Small Resnet on CIFAR-10, yielding an accuracy above 90%.
Revised: We train each possible Small Resnet on CIFAR-10, yielding an accuracy above 90%.

Original: We also generatea randomly initialized, untrained network for each Small Resnet architecture and conﬁrm that thesehave an accuracy of around 10% 2 .
Revised: We also generate a randomly initialized, untrained network for each Small Resnet architecture and conﬁrm that these have an accuracy of around 10% 3 .

Original: All these networks are frozen and cannot learn during stitching.
Revised: All these networks are frozen and cannot learn during stitching.

Original: We stitch every ordered pair of Small Resnets.
Revised: We stitch every ordered pair of Small Resnets.

Original: There are 16 · 16 = 256 such pairs.
Revised: There are 16 · 16 = 256 such pairs.

Original: In every orderedpair of networks being stitched, the former is called the sender , and the latter is the receiver .
Revised: In every ordered pair of networks being stitched, the former is called the sender , and the latter is the receiver .

Original: Astitch is used to transform the output of the sender at an intermediate layer before inputing it into anintermediate layer in the reciever.
Revised: A stitch is used to transform the output of the sender at an intermediate layer before inputing it into an intermediate layer in the reciever.

Original: For any network A we can consider layer i as A i , the ﬁrst i layers(assuming we start at zero) as A <i , and the layers after i as A
Revised: For any network A we can consider layer i as A i , the ﬁrst i layers (assuming we start at zero) as A <i , and the layers after i as A

Original: i< .
Revised: i< .

Original: If we wish to include layer i we canalways call such (partial) networks A ≤ i or A i ≤ .
Revised: If we wish to include layer i we can always call such (partial) networks A ≤ i or A i ≤ .

Original: For an input x , if we call the sender A , the reciever
Revised: For an input x , if we call the sender A , the reciever

Original: B , and the stitch S , we call C =
Revised: B , and the stitch S , we call C =

Original: B j< ( S ( A ≤ i ( x )))
Revised: B j< ( S ( A ≤ i ( x )))

Original: the stitched network .
Revised: the stitched network .

Original: Normally, we train S bydoing backpropagation on the stitched network with both A and B frozen.
Revised: Normally, we train S by doing backpropagation on the stitched network with both A and B frozen.

Original: The resulting accuracy isused to ﬁnd the similarity 3 between A ≤ i ( x ) and B ≤ j ( x ) , the former of which is called the providedrepresentation and the latter of which is called the expected representation .
Revised: The resulting accuracy is used to ﬁnd the similarity 4 between A ≤ i ( x ) and B ≤ j ( x ) , the former of which is called the provided representation and the latter of which is called the expected representation .

Original: Unlike Bansal et al., which only compare corresponding blocks (i.e. i = j ) of a sender and recieverwith the same architecture, we stitch from all residual blocks of the sender into all residual blocksof the receiver even when they have different architectures, as long as they are Small Resnets.
Revised: Unlike Bansal et al., which only compare corresponding blocks (i.e. i = j ) of a sender and reciever with the same architecture, we stitch from all residual blocks of the sender into all residual blocks of the receiver even when they have different architectures, as long as they are Small Resnets.

Original: Alsounlike Bansal et al.
Revised: Also unlike Bansal et al.

Original: , we only vary our neural networks by their initialization weights, but our setupis otherwise nearly identical.
Revised: , we only vary our neural networks by their initialization weights, but our setup is otherwise nearly identical.

Original: To be able to stitch between all blocks, we use strided convolutionsor upsampling when the dimensions are not the same.
Revised: To be able to stitch between all blocks, we use strided convolutions or upsampling when the dimensions are not the same.

Original: In our case the heights and widths vary bypowers of two so we can simple use 2x2, 4x4, or similarly-sized convolutions to downsample. 
Revised: In our case the heights and widths vary by powers of two so we can simple use 2x2, 4x4, or similarly-sized convolutions to downsample. For geomet-

Revised: There are ten classes in CIFAR-10.

Original: Forupsampling we use 2x, 4x, or similarly sized 2D nearest-upsampling (meaning that an element iscopied into a grid of equally-valued elements) before applying a 1x1 convolution in the stitch.
Revised: We use the downstream accuracy of the stitched network as our similarity upsampling we use 2x, 4x, or similarly sized 2D nearest-upsampling (meaning that an element is copied into a grid of equally-valued elements) before applying a 1x1 convolution in the stitch.

Original: We use the randomly-initialized ResNets as controls.
Revised: We use the randomly-initialized ResNets as controls.

Original: Our controls enable us to make sure that thestitches are appropriately powerful.
Revised: Our controls enable us to make sure that the stitches are appropriately powerful.

Original: By powerful we mean how complex the functions are that stitchescan represent.
Revised: By powerful we mean how complex the functions are that stitches can represent.

Original: If a stitch is very powerful, then even with random networks it should be able to yieldhigh downstream accuracy because it can learn any transformation.
Revised: If a stitch is very powerful, then even with random networks it should be able to yield high downstream accuracy because it can learn any transformation.

Original: In this case our stitches wouldalways yield high “similarity” and therefore be uninformative.
Revised: In this case our stitches would always yield high “similarity” and therefore be uninformative.

Original: We can be sure this is not the caseThere are We by ensuring that the stitches never yield high similarity for random networks (where only overlypowerful stitches would).
Revised: We can be sure this is not the case by ensuring that the stitches never yield high similarity for random networks (where only overly powerful stitches would).

Original:  Results
Revised:  Results

Original: For every ordered pair of networks, we plot the accuracy of all the stitched networks on a grid, basedon the sender’s layer and the receiver’s layer.
Revised: For every ordered pair of networks, we plot the accuracy of all the stitched networks on a grid, based on the sender’s layer and the receiver’s layer.

Original: The layer is denoted by an integer which counts howmany residual blocks came before it 4 .
Revised: The layer is denoted by an integer which counts how many residual blocks came before it 5 .

Original: The value in the grid element is the accuracy of the stitchednetwork after traning.
Revised: The value in the grid element is the accuracy of the stitched network after traning.

Original: Thus, this grid is a similarity matrix where entry i, j correspnds to the similarityof A ≤ i ( x ) and B ≤ j .
Revised: Thus, this grid is a similarity matrix where entry i, j correspnds to the similarity of A ≤ i ( x ) and B ≤ j .

Original: 3.1 Expectations
Revised: 3.1 Expectations

Original: We hypothesized that for similarity matrices between networks of the same architecture we would seea high similarity diagonal.
Revised: We hypothesized that for similarity matrices between networks of the same architecture we would see a high similarity diagonal.

Original: For networks of different architectures we hoped for a shorter diagonal(since the matrix isn’t square) or a diagonal with a different slope.
Revised: For networks of different architectures we hoped for a shorter diagonal (since the matrix isn’t square) or a diagonal with a different slope.

Original: Generally, however, we assumed that there would exist some non-negligible number (cid:15) such that ifwe mapped each sender layer’s representation to its most similar counterpart in the reciever, andtheir similarity was s , the similarity between that layer’s representation in the sender and every otherlayer’s in the reciever would be less than s − (cid:15) .
Revised: Generally, however, we assumed that there would exist some non-negligible number (cid:15) such that if we mapped each sender layer’s representation to its most similar counterpart in the reciever, and their similarity was s , the similarity between that layer’s representation in the sender and every other layer’s in the reciever would be less than s − (cid:15) .

Original: Moreover, we assumed that such a mapping wouldbe injective.
Revised: Moreover, we assumed that such a mapping would be injective.

Original: Intuitively, we thought that there would be a one-to-one correspondence between mostlayers in the sender to most layers in the reciever.
Revised: Intuitively, we thought that there would be a one-to-one correspondence between most layers in the sender to most layers in the reciever.

Original: We did not expect any layers’ representation in thesender to have high similarity to multiple layers’ representations in the reciever.
Revised: We did not expect any layers’ representation in the sender to have high similarity to multiple layers’ representations in the reciever.

Original: These hypotheses make sense because Bansal et.
Revised: These hypotheses make sense because Bansal et.

Original: al.
Revised: al.

Original: ’s
Revised: ’s

Original: ﬁndings suggest that each layer in the sendershould have at least one similar layer in the reciever—at least in the cast of identical architectures,where those two layers are the corresponding ones; and it is usually assumed that distant layers havedifferent information, and so they should not be similar.
Revised: ﬁndings suggest that each layer in the sender should have at least one similar layer in the reciever—at least in the cast of identical architectures, where those two layers are the corresponding ones; and it is usually assumed that distant layers have different information, and so they should not be similar.

Original: However, we found that every layer in thesender was extremely similar to all layers before it in the reciever .
Revised: However, we found that every layer in the sender was extremely similar to all layers before it in the reciever .

Original: That is to say, regardless ofthe architecture, if j ≤ i then the similarity was high.
Revised: That is to say, regardless of the architecture, if j ≤ i then the similarity was high.

Original: Visually, this looks like a triangle in the lowerleft-hand corner of the similarity matrix.
Revised: Visually, this looks like a triangle in the lower left-hand corner of the similarity matrix.

Original: This is visible in the ﬁgure below and quite perplexing.
Revised: This is visible in the ﬁgure below and quite perplexing.

Original: [Figure]

Original: For our controls we expected to see low stitching accuracy throughout the board since the networksare random, and we did.
Revised: For our controls we expected to see low stitching accuracy throughout the board since the networks are random, and we did.

Original: With that said, some of the top left or bottom right elements have highsimilarity depending on whether the sender or reciever was random.
Revised: With that said, some of the top left or bottom right elements have high similarity depending on whether the sender or reciever was random.

Original: In the case of a random sender[Figure] and trained reciever, if i and j are small, it is easy for S to undo A ≤ i , and give B j< something usable.[Figure]
Revised: In the case of a random sender and trained reciever, if i and j are small, it is easy for S to undo A ≤ i , and give B j< something usable.

Original: The opposite case is analogous.
Revised: The opposite case is analogous.

Original: [Figure]

Original: [Figure]

Original:  Conclusion
Revised: 3.2 Conclusion

Original: The most interesting aspect of our results is the high accuracy of the stitching network for layers inthe lower left hand triangle.
Revised: The most interesting aspect of our results is the high accuracy of the stitching network for layers in the lower left hand triangle.

Revised: It is true that our ﬁndings do not contradict Bansal et.

Revised: al.

Revised: ’s

Revised: ﬁndings since they only stitched on the diagonal which yielded the same high accuracy for them as it did for us.

Revised: However, we are still surprised.

Original: Given that we interpret the stitching network accuracy as a similarity, ourresults suggest that each sender representation is similar with all the receiver representations from a layer before it.
Revised: Given that we interpret the stitching network accuracy as a similarity, our results suggest that each sender representation is similar with all the receiver representations from a layer before it.

Original: We expected to see that each layer would be similar to a couple (nearby) layers at mostbecause the standard narrative has been that every layer loses some amount of granular information,and so that information should not be reconstructable in an interchangeability test like stitching.
Revised: We expected to see that each layer would be similar to a couple (nearby) layers at most because the standard narrative has been that every layer loses some amount of granular information, and so that information should not be reconstructable in an interchangeability test like stitching.

Original: We see two main explanations for the results.
Revised: We see two main explanations for the results.

Original: The ﬁrst is that the common narrative could be wrongand some neural networks may in fact be able to maintain most if not all of the granular informationof the image throughout their processing of it.
Revised: The ﬁrst is that the common narrative could be wrong and some neural networks may in fact be able to maintain most if not all of the granular information of the image throughout their processing of it.

Original: The second is that the stitch may be able to give thereciever a representation which, despite being different from that which is expected, nonethelessyields high accuracy.
Revised: The second is that the stitch may be able to give the reciever a representation which, despite being different from that which is expected, nonetheless yields high accuracy.

Original: We believe that the latter option is the most likely.
Revised: We bet that the latter option is the most likely.

Original: Intuitively, the stitch may be able to ﬁgure out howto generate some generic, albeit unrealistic, set of the most salient features for the recieving layerto classify in a given way.
Revised: Intuitively, the stitch may be able to ﬁgure out how to generate some generic, albeit unrealistic, set of the most salient features for the recieving layer to classify in a given way.

Original: Perhaps it is generating the “average” human, for example.
Revised: Perhaps it is generating the “average” human, for example.

Original: However, adeeper analysis is required to ascertain whether that is the case.
Revised: However, a deeper analysis is required to ascertain whether that is the case.

Original: In the Appendix, we further discusssanity tests we explored and dub this hypothesis hacking .
Revised: In the Appendix, we further discuss sanity tests we explored and dub this hypothesis hacking .

Original: Despite our difﬁculty generalizing model stitching, we still see it as an important step forward inour ability to compare representations, since its focus on functional similarity makes its resultsmore salient than those from geometric closeness or statistical measures.
Revised: Despite our difﬁculty generalizing model stitching, we still see it as an important step forward in our ability to compare representations, since its focus on functional similarity makes its results more salient than those from geometric closeness or statistical measures.

Original: Unlike existing measuresof similarity which tend to look for literal distance between representations, stitching follows aprocess whereby we deﬁne what behaviors should be exhibited by similar representations (i.e. theyshould be interchangeable up to a degree of ﬂexibility determined by the function class of the stitch)and then devise tasks/experiments that test these behaviors on the representations under question.
Revised: Unlike existing measures of similarity which tend to look for literal distance between representations, stitching follows a process whereby we deﬁne what behaviors should be exhibited by similar representations (i.e. they should be interchangeable up to a degree of ﬂexibility determined by the function class of the stitch) and then devise tasks/experiments that test these behaviors on the representations under question.

Original: This approach is far better, because the research community understands tasks much better than thenumerical, geometric, or statistical properties of representations.
Revised: This approach is far better, because the research community understands tasks much better than the numerical, geometric, or statistical properties of representations.

Original: Moreover, it is easier to interpret theresulting similarity measurements in terms of accuracy or other such functional quantities, makingtechniques like stitching more useful, even in practice.
Revised: Moreover, it is easier to interpret the resulting similarity measurements in terms of accuracy or other such functional quantities, making techniques like stitching more useful, even in practice.

Original: We hope to see more representationalcomparison techniques following the high-level process outlined above in the future.
Revised: We hope to see more representational comparison techniques following the high-level process outlined above in the future

Revised: Acknowledgments and Disclosure of Funding

Revised: Thank you to MIT EECS for funding this project and MIT SuperUROP and the Soljacic Lab for supporting it.

Original: A Appendix
Revised: A Appendix

Original: Here we provide additional information that may be helpful to readers, but which was beyond the scope of the155 main paper.
Revised: Here we provide additional information that may be helpful to readers, but which was beyond the scope of the main paper.

Original: There are four subsections: further details on our testing procedures; results from larger Resnetsto support the idea that our surprising ﬁndings could generalize; numerical sanity testing to try and conﬁrm157 or disprove the hacking hypothesis; and image-generation to try and visualize the possibility of the hackinghypothesis.
Revised: There are four subsections: further details on our testing procedures; results from larger Resnets to support the idea that our surprising ﬁndings could generalize; numerical sanity testing to try and conﬁrm or disprove the hacking hypothesis; and image-generation to try and visualize the possibility of the hacking hypothesis.

Original: Experimental Details
Revised: Experimental Details

Original: We train our stitches for four epochs with momentum 0.9, batch size 256, weight decay 0.01, learning rate 0.01,161 and a post-warmup cosine learning rate scheduler.
Revised: We train our stitches for four epochs with momentum 0.9, batch size 256, weight decay 0.01, learning rate 0.01, and a post-warmup cosine learning rate scheduler.

Original: We chose our hyperparameters because they were effectivefor training the Small Resnets between which we stitched.
Revised: We chose our hyperparameters because they were effective for training the Small Resnets between which we stitched.

Original: Below is an example of a Small Resnet for clarity on163 our architecture.
Revised: Below is an example of a Small Resnet for clarity on our architecture.

Original: Numerical Sanity Testing
Revised: Numerical Sanity Testing

Original: When we found that our metric was ﬁnding high similarity between distant layers, we decided to sanity test169 this result using numerical testing.
Revised: When we found that our metric was ﬁnding high similarity between distant layers, we decided to sanity test this result using numerical testing.

Original: We averaged the mean squared error between three pairs of values: theexpected representation with that generated by a vanilla stitch (trained with backpropagation on the CIFAR-10171 classiﬁcation task, as discussed in the body of this paper); the expected representation with that generated by asimilarity-trained stitch; and the representation generated by similarity-trained stitch with that generated by a173 vanilla stitch.
Revised: We averaged the mean squared error between three pairs of values: the expected representation with that generated by a vanilla stitch (trained with backpropagation on the CIFAR-10 classiﬁcation task, as discussed in the body of this paper); the expected representation with that generated by a similarity-trained stitch; and the representation generated by similarity-trained stitch with that generated by a vanilla stitch.

Original: Respectively, we refer to these three pairs as EV , ES , and SV .
Revised: Respectively, we refer to these three pairs as EV , ES , and SV .

Original: The mean squared error is overthe elements in the representations’ tensors.
Revised: The mean squared error is over the elements in the representations’ tensors.

Original: For those mean squared errors, wefound the minimum, mean,175 maximum, and standard deviation over the (cartesian product of the) entire datasetof CIFAR-10 and all the pairsof layersacross all Small Resnets.
Revised: For those mean squared errors, we the minimum, mean, maximum, and standard deviation over the (cartesian product of the entire dataset CIFAR-10 and all the pairs of layers all Small Resnets.

Original: We alsomeasured thesame statistics for only corresponding layers (i.e. layer177 one with layer one) to get a baseline similar to what Bansal et.
Revised: also the statistics for only corresponding layers (i.e layer one with layer one) to get a baseline similar to what Bansal et.

Original: al.
Revised: al.

Original: ’swork may have yielded.
Revised: ’s may have yielded.

Original: Unlike the vanilla stitch, the similarity-trained stitch was trained to minimize the mean squared error between179 the expected representation of the reciever (at a layer) and the output of the corresponding stitch.
Revised: Unlike the vanilla stitch, the similarity-trained stitch was trained to minimize the mean squared error between the expected representation of the reciever (at a layer) and the output of the corresponding stitch.

Original: For example,consider the input x , sender A , reciever B , and stitch S .
Revised: For example, consider the input x , sender A , reciever B , and stitch S .

Original: Consider the stitched network C =
Revised: Consider the stitched network C =

Original: B j< ( S ( A ≤ i )) , and181 recall that the expected representation is B ≤ j (that is, the computation up to, including, layer j of the reciever).
Revised: B j< ( S ( A ≤ i )) , and recall that the expected representation is B ≤ j (that is, the computation up to, including, layer j of the reciever).

Original: The vanilla stitch would be trained using backpropagation on C with all the weights of A and B frozen (only theweights in S can change).
Revised: The vanilla stitch would be trained using backpropagation on C with all the weights of A and B frozen (only the weights in S can change).

Original: However, the similarity-trained stitch would be trained on ( S ( A <i ) − B ≤ j ) 2 .
Revised: However, the similarity-trained stitch would be trained on ( S ( A <i ) − B ≤ j ) 2 .

Original: The184 purpose of the similarity-trained stitch is to get a baseline for what a “low” mean squared error is.
Revised: The purpose of the similarity-trained stitch is to get a baseline for what a “low” mean squared error is.

Original: We include itstask accuracy at the end of this section as a curiosity.
Revised: We include its task accuracy at the end of this section as a curiosity.

Original: Below we plot our results in two tables.
Revised: Below we plot our results in two tables.

Original: We denote the highest difference with red, that with the second highest187 difference with yellow, and that with the lowest difference with green.
Revised: We denote the highest difference with red, that with the second highest difference with yellow, and that with the lowest difference with green.

Original: We expect, therefore, to see red in the
Revised: We expect, therefore, to see red in the

Original: EV and SV columns and green in the ES column.
Revised: EV and SV columns and green in the ES column.

Original: That is because the similarity-trained stitch should be closer189 to the expected representation (since the their difference is its loss function) than the vanilla stitch is to either.
Revised: That is because the similarity-trained stitch should be closer to the expected representation (since the their difference is its loss function) than the vanilla stitch is to either.

Original: If the vanilla stitch is learning to hack the reciever then we expect its difference to be larger by many orders191 of magnitude than the similarity-trained stitch.
Revised: If the vanilla stitch is learning to hack the reciever then we expect its difference to be larger by many orders of magnitude than the similarity-trained stitch.

Original: While we do see that our prediction is typically correct, thedifference is not as large nor as decimatingly common as we had hoped, and so we cannot conclude, from these193 results, that the stitch is likely hacking the reciever.
Revised: While we do see that our prediction is typically correct, the difference is not as large nor as decimatingly common as we had hoped, and so we cannot conclude, from these results, that the stitch is likely hacking the reciever.

Original: [Table]
Revised: [Table]

Original: Stitches

Original: [Table]
Revised: [Table]