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The Grand Tour*tasks* a high-dimensional dataset into two dimensions.

Over time, the Grand Tour easily animates its projection so that each attainable view of the dataset is (finally) introduced to the viewer.

In contrast to trendy nonlinear projection strategies similar to t-SNE*linear* methodology.

On this article, we present how one can leverage the linearity of the Grand Tour to allow a lot of capabilities which are uniquely helpful to visualise the conduct of neural networks.

Concretely, we current three use circumstances of curiosity: visualizing the coaching course of because the community weights change, visualizing the layer-to-layer conduct as the info goes by means of the community and visualizing each how adversarial examples

## Introduction

Deep neural networks usually obtain best-in-class efficiency in supervised studying contests such because the ImageNet Massive Scale Visible Recognition Problem (ILSVRC)

Sadly, their determination course of is notoriously arduous to interpret

On this article, we current a way to visualise the responses of a neural community which leverages properties of deep neural networks and properties of the *Grand Tour*.

Notably, our methodology allows us to extra immediately motive in regards to the relationship between *adjustments within the information* and *adjustments within the ensuing visualization*

As we are going to present, this data-visual correspondence is central to the strategy we current, particularly when in comparison with different non-linear projection strategies like UMAP and t-SNE.

To grasp a neural community, we regularly attempt to observe its motion on enter examples (each actual and synthesized)

These sorts of visualizations are helpful to elucidate the activation patterns of a neural community for a single instance, however they could supply much less perception in regards to the relationship between totally different examples, totally different states of the community because it’s being skilled, or how the info within the instance flows by means of the totally different layers of a single community.

Subsequently, we as an alternative goal to allow visualizations of the *context round* our objects of curiosity: what’s the distinction between the current coaching epoch and the following one? How does the classification of a community converge (or diverge) because the picture is fed by means of the community?

Linear strategies are engaging as a result of they’re notably straightforward to motive about.

The Grand Tour works by producing a random, easily altering rotation of the dataset, after which projecting the info to the two-dimensional display screen: each are linear processes.

Though deep neural networks are clearly not linear processes, they usually confine their nonlinearity to a small set of operations, enabling us to nonetheless motive about their conduct.

Our proposed methodology higher preserves context by offering extra

consistency: it needs to be attainable to know *how the visualization
would change, if the info had been totally different in a selected
approach*.

## Working Examples

For example the approach we are going to current, we skilled deep neural

community fashions (DNNs) with 3 widespread picture classification datasets:

MNIST

MNIST

Picture credit score to https://en.wikipedia.org/wiki/File:MnistExamples.png

fashion-MNIST

Style-MNIST

Picture credit score to https://towardsdatascience.com/multi-label-classification-and-class-activation-map-on-fashion-mnist-1454f09f5925

and CIFAR-10

CIFAR-10

Picture credit score to https://www.cs.toronto.edu/~kriz/cifar.html

Whereas our structure is less complicated and smaller than present DNNs, it’s nonetheless indicative of recent networks, and is advanced sufficient to exhibit each our proposed methods and shortcomings of typical approaches.

The next determine presents a easy practical diagram of the neural community we are going to use all through the article. The neural community is a sequence of linear (each convolutional

A convolution calculates weighted sums of areas within the enter.

In neural networks, the learnable weights in convolutional layers are known as the kernel.

For instance

Picture credit score to https://towardsdatascience.com/gentle-dive-into-math-behind-convolutional-neural-networks-79a07dd44cf9.

See additionally Convolution arithmetic.

A completely-connected layer computes output neurons as weighted sum of enter neurons. In matrix kind, it’s a matrix that linearly transforms the enter vector into the output vector.

First launched by Nair and Hinton

Softmax operate calculates $S(y_i)=frac{e^{y_i}}{Sigma_{j=1}^{N} e^{y_j}}$

Though neural networks are able to unbelievable feats of classification, deep down, they are surely simply pipelines of comparatively easy features. For photos, the enter is a 2D array of scalar values for grey scale photos or RGB triples for coloured photos. When wanted, one can at all times flatten the 2D array into an equal ($w cdot h cdot c$

Many of the easy features fall into two classes: they’re both linear transformations of their inputs (like fully-connected layers or convolutional layers), or comparatively easy non-linear features that work component-wise (like sigmoid activations

The above determine helps us have a look at a single picture at a time; nonetheless, it doesn’t present a lot context to know the connection between layers, between totally different examples, or between totally different class labels. For that, researchers usually flip to extra refined visualizations.

## Utilizing Visualization to Perceive DNNs

Let’s begin by contemplating the issue of visualizing the coaching technique of a DNN. When coaching neural networks, we optimize parameters within the operate to attenuate a scalar-valued loss operate, usually by means of some type of gradient descent. We would like the loss to maintain lowering, so we monitor the entire historical past of coaching and testing losses over rounds of coaching (or “epochs”), to guarantee that the loss decreases over time. The next determine exhibits a line plot of the coaching loss for the MNIST classifier.

Though its basic development meets our expectation because the loss steadily decreases, we see one thing unusual round epochs 14 and 21: the curve goes nearly flat earlier than beginning to drop once more. What occurred? What precipitated that?

If we separate enter examples by their true labels/lessons and plot the *per-class* loss like above, we see that the 2 drops had been brought on by the classses 1 and seven; the mannequin learns totally different lessons at very totally different instances within the coaching course of.
Though the community learns to acknowledge digits 0, 2, 3, 4, 5, 6, 8 and 9 early on, it’s not till epoch 14 that it begins efficiently recognizing digit 1, or till epoch 21 that it acknowledges digit 7.
If we knew forward of time to be in search of class-specific error charges, then this chart works properly. However what if we didn’t actually know what to search for?

In that case, we may take into account visualizations of neuron activations (e.g. within the final softmax layer) for *all* examples without delay, trying
to search out patterns like class-specific conduct, and different patterns apart from.
Ought to there be solely two neurons in that layer, a easy two-dimensional scatter plot would work.
Nonetheless, the factors within the softmax layer for our instance datasets are 10 dimensional (and in larger-scale classification issues this quantity might be a lot bigger).
We have to both present two dimensions at a time (which doesn’t scale properly because the variety of attainable charts grows quadratically),
or we are able to use *dimensionality discount* to map the info right into a two dimensional area and present them in a single plot.

### The State-of-the-art Dimensionality Discount is Non-linear

Fashionable dimensionality discount methods similar to t-SNE and UMAP are able to spectacular feats of summarization, offering two-dimensional photos the place comparable factors are usually clustered collectively very successfully.
Nonetheless, these strategies aren’t notably good to know the conduct of neuron activations at a advantageous scale.
Contemplate the aforementioned intriguing characteristic in regards to the totally different studying price that the MNIST classifier has on digit 1 and seven: the community didn’t study to acknowledge digit 1 till epoch 14, digit 7 till epoch 21.
We compute t-SNE, Dynamic t-SNE

One motive that non-linear embeddings fail in elucidating this phenomenon is that, for the actual change within the information, the fail the precept of *data-visual correspondence* *match in magnitude*: a barely noticeable change in visualization needs to be as a result of smallest attainable change in information, and a salient change in visualization ought to replicate a major one in information.
Right here, a major change occurred in solely a *subset* of knowledge (e.g. all factors of digit 1 from epoch 13 to 14), however *all* factors within the visualization transfer dramatically.
For each UMAP and t-SNE, the place of every single level relies upon non-trivially on the entire information distribution in such embedding algorithms.
This property will not be perfect for visualization as a result of it fails the data-visual correspondence, making it arduous to *infer* the underlying change in information from the change within the visualization.

Non-linear embeddings which have non-convex aims additionally are usually delicate to preliminary situations.
For instance, in MNIST, though the neural community begins to stabilize on epoch 30, t-SNE and UMAP nonetheless generate fairly totally different projections between epochs 30, 31 and 32 (the truth is, all the way in which to 99).
Temporal regularization methods (similar to Dynamic t-SNE) mitigate these consistency points, however nonetheless endure from different interpretability points

Now, let’s take into account one other process, that of figuring out lessons which the neural community tends to confuse. For this instance, we are going to use the Style-MNIST dataset and classifier, and take into account the confusion amongst sandals, sneakers and ankle boots. If we all know forward of time that these three lessons are more likely to confuse the classifier, then we are able to immediately design an acceptable linear projection, as might be seen within the final row of the next determine (we discovered this specific projection utilizing each the Grand Tour and the direct manipulation approach we later describe). The sample on this case is kind of salient, forming a triangle. T-SNE, in distinction, incorrectly separates the category clusters (presumably due to an inappropriately-chosen hyperparameter). UMAP efficiently isolates the three lessons, however even on this case it’s not attainable to tell apart between three-way confusion for the classifier in epochs 5 and 10 (portrayed in a linear methodology by the presence of factors close to the middle of the triangle), and a number of two-way confusions in later epochs (evidences by an “empty” heart).

## Linear Strategies to the Rescue

When given the possibility, then, we should always desire strategies for which adjustments within the information produce predictable, visually salient adjustments within the end result, and linear dimensionality reductions usually have this property. Right here, we revisit the linear projections described above in an interface the place the person can simply navigate between totally different coaching epochs. As well as, we introduce one other helpful functionality which is barely accessible to linear strategies, that of direct manipulation. Every linear projection from $n$

This setup gives extra good properties that specify the salient patterns within the earlier illustrations. For instance, as a result of projections are linear and the coefficients of vectors within the classification layer sum to 1, classification outputs which are midway assured between two lessons are projected to vectors which are midway between the category handles.

This specific property is illustrated clearly within the Style-MNIST instance beneath. The mannequin confuses sandals, sneakers and ankle boots, as information factors kind a triangular form within the softmax layer.

Examples falling between lessons point out that the mannequin has hassle distinguishing the 2, similar to sandals vs. sneakers, and sneakers vs. ankle boot lessons. Notice, nonetheless, that this doesn’t occur as a lot for sandals vs. ankle boots: not many examples fall between these two lessons. Furthermore, most information factors are projected near the sting of the triangle. This tells us that the majority confusions occur between two out of the three lessons, they’re actually two-way confusions.

Throughout the similar dataset, we are able to additionally see pullovers, coats and shirts filling a triangular *airplane*.
That is totally different from the sandal-sneaker-ankle-boot case, as examples not solely fall on the boundary of a triangle, but in addition in its inside: a real three-way confusion.

Equally, within the CIFAR-10 dataset we are able to see confusion between canine and cats, airplanes and ships. The blending sample in CIFAR-10 will not be as clear as in fashion-MNIST, as a result of many extra examples are misclassified.

## The Grand Tour

Within the earlier part, we took benefit of the truth that we knew which lessons to visualise.
That meant it was straightforward to design linear projections for the actual duties at hand.
However what if we don’t know forward of time which projection to select from, as a result of we don’t fairly know what to search for?
Principal Part Evaluation (PCA) is the quintessential linear dimensionality discount methodology,
selecting to undertaking the info in order to protect essentially the most variance attainable.
Nonetheless, the distribution of knowledge in softmax layers usually has comparable variance alongside many axis instructions, as a result of every axis concentrates an identical variety of examples across the class vector.

Beginning with a random velocity, it easily rotates information factors across the origin in excessive dimensional area, after which tasks it all the way down to 2D for show. Listed below are some examples of how Grand Tour acts on some (low-dimensional) objects:

- On a sq., the Grand Tour rotates it with a relentless angular velocity.
- On a dice, the Grand Tour rotates it in 3D, and its 2D projection allow us to see each aspect of the dice.
- On a 4D dice (a
*tesseract*), the rotation occurs in 4D and the 2D view exhibits each attainable projection.

### The Grand Tour of the Softmax Layer

We first have a look at the Grand Tour of the softmax layer. The softmax layer is comparatively straightforward to know as a result of its axes have sturdy semantics. As we described earlier, the $i$

^{th}) epoch, with MNIST, fashion-MNIST or CIFAR-10 dataset.

The Grand Tour of the softmax layer lets us qualitatively assess the efficiency of our mannequin.
Within the specific case of this text, since we used comparable architectures for 3 datasets, this additionally permits us to gauge the relative problem of classifying every dataset.
We are able to see that information factors are most confidently labeled for the MNIST dataset, the place the digits are near one of many ten corners of the softmax area. For Style-MNIST or CIFAR-10, the separation will not be as clear, and extra factors seem *inside* the quantity.

### The Grand Tour of Coaching Dynamics

Linear projection strategies naturally give a formulation that’s impartial of the enter factors, permitting us to maintain the projection mounted whereas the information adjustments. To recap our working instance, we skilled every of the neural networks for 99 epochs and recorded the complete historical past of neuron activations on a subset of coaching and testing examples. We are able to use the Grand Tour, then, to visualise the precise coaching course of of those networks.

At first when the neural networks are randomly initialized, all examples are positioned across the heart of the softmax area, with equal weights to every class.
Via coaching, examples transfer to class vectors within the softmax area. The Grand Tour additionally lets us
examine visualizations of the coaching and testing information, giving us a qualitative evaluation of over-fitting.
Within the MNIST dataset, the trajectory of testing photos by means of coaching is in line with the coaching set.
Knowledge factors went immediately towards the nook of its true class and all lessons are stabilized after about 50 epochs.
Alternatively, in CIFAR-10 there may be an *inconsistency* between the coaching and testing units. Photographs from the testing set maintain oscillating whereas most photos from coaching converges to the corresponding class nook.
In epoch 99, we are able to clearly see a distinction in distribution between these two units.
This alerts that the mannequin overfits the coaching set and thus doesn’t generalize properly to the testing set.

### The Grand Tour of Layer Dynamics

Given the introduced methods of the Grand Tour and direct manipulations on the axes, we are able to in concept visualize and manipulate any intermediate layer of a neural community by itself. Nonetheless, this isn’t a really satisfying method, for 2 causes:

- In the identical approach that we’ve saved the projection mounted because the coaching information modified, we want to “maintain the projection mounted”, as the info strikes by means of the layers within the neural community. Nonetheless, this isn’t simple. For instance, totally different layers in a neural community have totally different dimensions. How can we join rotations of 1 layer to rotations of the opposite?
- The category “axis handles” within the softmax layer handy, however that’s solely sensible when the dimensionality of the layer is comparatively small. With lots of of dimensions, for instance, there could be too many axis handles to naturally work together with. As well as, hidden layers would not have as clear semantics because the softmax layer, so manipulating them wouldn’t be as intuitive.

To deal with the primary drawback, we might want to pay nearer consideration to the way in which by which layers remodel the info that they’re given. To see how a linear transformation might be visualized in a very ineffective approach, take into account the next (quite simple) weights (represented by a matrix $A$

$x_t = (1-t) cdot x_0 + t cdot x_1 = (1-2t) cdot x_0$

Successfully, this technique reuses the linear projection coefficients from one layer to the following. It is a pure thought, since they’ve the identical dimension. Nonetheless, discover the next: the transformation given by A is an easy rotation of the info. Each linear transformation of the layer $ok+1$

This commentary factors to a extra basic technique: when designing a visualization, we needs to be as express as attainable about which components of the enter (or course of) we search to seize in our visualizations.
We should always search to explicitly articulate what are purely representational artifacts that we should always discard, and what are the true includes a visualization we should always *distill* from the illustration.
Right here, we declare that rotational elements in linear transformations of neural networks are considerably much less necessary than different elements similar to scalings and nonlinearities.
As we are going to present, the Grand Tour is especially engaging on this case as a result of it’s might be made to be invariant to rotations in information.
Consequently, the rotational elements within the linear transformations of a neural community can be explicitly made invisible.

Concretely, we obtain this by profiting from a central theorem of linear algebra.
The *Singular Worth Decomposition* (SVD) theorem exhibits that *any* linear transformation might be decomposed right into a sequence of quite simple operations: a rotation, a scaling, and one other rotation

Making use of a matrix $A$

(For the next portion, we scale back the variety of information factors to 500 and epochs to 50, with a view to scale back the quantity of knowledge transmitted in a web-based demonstration.) With the linear algebra construction at hand, now we’re in a position to hint behaviors and patterns from the softmax again to earlier layers. In fashion-MNIST, for instance, we observe a separation of footwear (sandals, sneakers and ankle boots as a gaggle) from all different lessons within the softmax layer. Tracing it again to earlier layers, we are able to see that this separation occurred as early as layer 5:

### The Grand Tour of Adversarial Dynamics

As a last software situation, we present how the Grand Tour can even elucidate the conduct of adversarial examples

Via this adversarial coaching, the community finally claims, with excessive confidence, that the inputs given are all 0s. If we keep within the softmax layer and slide although the adversarial coaching steps within the plot, we are able to see adversarial examples transfer from a excessive rating for sophistication 8 to a excessive rating for sophistication 0. Though all adversarial examples are labeled because the goal class (digit 0s) finally, a few of them detoured someplace near the centroid of the area (across the twenty fifth epoch) after which moved in the direction of the goal. Evaluating the precise photos of the 2 teams, we see those who these “detouring” photos are usually noisier.

Extra fascinating, nonetheless, is what occurs within the intermediate layers. In pre-softmax, for instance, we see that these faux 0s behave in another way from the real 0s: they stay nearer to the choice boundary of two lessons and kind a airplane by themselves.

## Dialogue

### Limitations of the Grand Tour

Early on, we in contrast a number of state-of-the-art dimensionality discount methods with the Grand Tour, exhibiting that non-linear strategies would not have as many fascinating properties because the Grand Tour for understanding the conduct of neural networks. Nonetheless, the state-of-the-art non-linear strategies include their very own energy. Each time geometry is worried, just like the case of understanding multi-way confusions within the softmax layer, linear strategies are extra interpretable as a result of they protect sure geometrical buildings of knowledge within the projection. When topology is the primary focus, similar to once we need to cluster the info or we’d like dimensionality discount for downstream fashions which are much less delicate to geometry, we’d select non-linear strategies similar to UMAP or t-SNE for they’ve extra freedom in projecting the info, and can usually make higher use of the less dimensions accessible.

### The Energy of Animation and Direct Manipulation

When evaluating linear projections with non-linear dimensionality reductions, we used small multiples to distinction coaching epochs and dimensionality discount strategies.
The Grand Tour, alternatively, makes use of a single animated view.
When evaluating small multiples and animations, there isn’t a basic consensus on which one is healthier than the opposite within the literature, apart.
from particular settings similar to dynamic graph drawing

### Non-sequential Fashions

In our work we’ve used fashions which are purely “sequential”, within the sense that the layers might be put in numerical ordering, and that the activations for the $n+1$

### Scaling to Bigger Fashions

Fashionable architectures are additionally vast. Particularly when convolutional layers are involved, one may run into points with scalability if we see such layers as a big sparse matrix appearing on flattened multi-channel photos.
For the sake of simplicity, on this article we brute-forced the computation of the alignment of such convolutional layers by writing out their express matrix illustration.
Nonetheless, the singular worth decomposition of multi-channel 2D convolutions might be computed effectively

## Technical Particulars

This part presents the technical particulars essential to implement the direct manipulation of axis handles and information factors, in addition to how one can implement the projection consistency approach for layer transitions.

### Notation

On this part, our notational conference is that information factors are represented as row vectors. A complete dataset is laid out as a matrix, the place every row is an information level, and every column represents a special characteristic/dimension. Consequently, when a linear transformation is utilized to the info, the row vectors (and the info matrix general) are left-multiplied by the transformation matrix. This has a aspect profit that when making use of matrix multiplications in a sequence, the system reads from left to proper and aligns with a commutative diagram. For instance, when an information matrix $X$

$X overset{M}{mapsto} Y$

Moreover, if the SVD of $M$

### Direct Manipulation

The direct manipulations we introduced earlier present express management over the attainable projections for the info factors. We offer two modes: immediately manipulating class axes (the “axis mode”), or immediately manipulating a gaggle of knowledge factors by means of their centroid (the “information level mode”). Primarily based on the dimensionality and axis semantics, as mentioned in Layer Dynamics, we might desire one mode than the opposite.

We’ll see that the axis mode is a particular case of knowledge level mode, as a result of we are able to view an axis deal with as a selected “fictitious” level within the dataset. Due to its simplicity, we are going to first introduce the axis mode.

#### The Axis Mode

The implied semantics of direct manipulation is that when a person drags an UI component (on this case, an axis deal with), they’re signaling to the system that they wished that the corresponding information level had been projected to the situation the place the UI component was dropped, somewhat than the place it was dragged from. In our case the general projection is a rotation (initially decided by the Grand Tour), and an arbitrary person manipulation won’t essentially generate a brand new projection that can also be a rotation. Our purpose, then, is to discover a new rotation which satisfies the person request and is near the earlier state of the Grand Tour projection, in order that the ensuing state satisfies the person request.

In a nutshell, when person drags the $i^{th}$

Earlier than we see intimately why this works properly, allow us to formalize the method of the Grand Tour on a typical foundation vector $e_i$

$e_i overset{GT}{mapsto} tilde{e_i} overset{pi_2}{mapsto} (x_i, y_i)$

When person drags an axis deal with on the display screen canvas, they induce a delta change $Delta = (dx, dy)$

To discover a close by Grand Tour rotation that respects this transformation, first observe that $tilde{e_i}$

####
The Knowledge Level Mode

We now clarify how we immediately manipulate information factors.

Technically talking, this methodology solely considers one level at a time.

For a gaggle of factors, we compute their centroid and immediately manipulate this single level with this methodology.

Considering extra fastidiously in regards to the course of in axis mode offers us a method to drag any single level.

Recall that in axis mode, we added person’s manipulation $tilde{Delta} := (dx, dy, 0, 0, cdots)$

Wanting on the geometry of this motion, the “add-delta-then-normalize” on $tilde{e_i}$

The determine exhibits the case in 3D, however in larger dimensional area it’s basically the identical, for the reason that two vectors $tilde{e_i}$

Generalizing this commentary from axis deal with to arbitrary information level, we need to discover the rotation that strikes the centroid of a specific subset of knowledge factors $tilde{c}$

First, the angle of rotation might be discovered by their cosine similarity:

$theta = textrm{arccos}(
frac{langle tilde{c}, tilde{c}^{(new)} rangle}{||tilde{c}|| cdot ||tilde{c}^{(new)}||}
)$

$Q := start{bmatrix} cdots textsf{normalize}(tilde{c}) cdots cdots textsf{normalize}(tilde{c}^{(new)}_{perp}) cdots P finish{bmatrix}$

Making use of $Q$

###

Layer Transitions

####
ReLU Layers

####
Linear Layers

If $X^{l} = X^{l-1} M$

####
Convolutional Layers

With a change of illustration, we are able to animate a convolutional layer just like the earlier part.

For 2D convolutions this transformation of illustration includes flattening the enter and output, and repeating the kernel sample in a sparse matrix $M in mathbb{R}^{m instances n}$

####
Max-pooling Layers

We substitute it by average-pooling and scaling by the ratio of the common to the max.

We compute the matrix type of average-pooling and use its SVD to align the view earlier than and after this layer.

Functionally, our operations have equal outcomes to max-pooling, however this introduces

surprising artifacts. For instance, the max-pooling model of the vector $[0.9, 0.9, 0.9, 1.0]$

## Conclusion

As highly effective as t-SNE and UMAP are, they usually fail to supply the correspondences we’d like, and such correspondences can come, surprisingly, from comparatively easy strategies just like the Grand Tour. The Grand Tour methodology we introduced is especially helpful when direct manipulation from the person is accessible or fascinating.

We imagine that it is likely to be attainable to design strategies that spotlight the very best of each worlds, utilizing non-linear dimensionality discount to create intermediate, comparatively low-dimensional representations of the activation layers, and utilizing the Grand Tour and direct manipulation to compute the ultimate projection.

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