SDBM: Supervised Decision Boundary Maps for Machine Learning

Classiﬁers

Artur Andr

e A. M. Oliveira

1 a

, Mateus Espadoto

1 b

, Roberto Hirata Jr.

1 c

and Alexandru C. Telea

2 d

Institute of Mathematics and Statistics, University of S

ao Paulo, Brazil

Department of Information and Computing Sciences, Utrecht University, The Netherlands

Keywords:

Machine Learning, Dimensionality Reduction, Dense Maps.

Abstract:

Understanding the decision boundaries of a machine learning classiﬁer is key to gain insight on how classiﬁers

work. Recently, a technique called Decision Boundary Map (DBM) was developed to enable the visualization of

such boundaries by leveraging direct and inverse projections. However, DBM have scalability issues for creating

ﬁne-grained maps, and can generate results that are hard to interpret when the classiﬁcation problem has many

classes. In this paper we propose a new technique called Supervised Decision Boundary Maps (SDBM), which

uses a supervised, GPU-accelerated projection technique that solves the original DBM shortcomings. We show

through several experiments that SDBM generates results that are much easier to interpret when compared to

DBM, is faster and easier to use, while still being generic enough to be used with any type of single-output

classiﬁer.

1 INTRODUCTION

In recent years, Machine Learning (ML) techniques

have become very popular in many ﬁelds to support

pattern recognition and predictive modelling. Despite

their popularity, the inner workings of trained ML

models are hard to explain, which can hamper their

adoption where transparency and accountability of

inference is required (Ribeiro et al., 2016). For Deep

Learning (DL) models, explainability is an even harder

concern, as such models have millions of parameters

that contribute jointly to the generation of many levels

of latent features (Garcia et al., 2018).

For the more speciﬁc case of ML classiﬁers, sev-

eral approaches for model explanation have been pro-

posed, using variable importance (Lundberg and Lee,

2017), locally interpretable models (Ribeiro et al.,

2016), and a variety of visualization-based tech-

niques (Rauber et al., 2017b; Rauber et al., 2017a).

Garcia et al. (Garcia et al., 2018) recently presented a

survey of visual techniques oriented towards the expla-

nation of DL models.

https://orcid.org/0000-0002-3606-1687

https://orcid.org/0000-0002-1922-4309

https://orcid.org/0000-0003-3861-7260

https://orcid.org/0000-0003-0750-0502

A particular visual explanation technique in this

set is the Decision Boundary Map (DBM) (Rodrigues

et al., 2019). DBM extends classical multidimensional

projections (Nonato and Aupetit, 2018) by ﬁlling in the

gaps between projected points from a labeled dataset

used to train a classiﬁer with synthesized, classiﬁed,

data points. This effectively creates a 2D dense im-

age that shows how the classiﬁer partitions its high-

dimensional data space into per-class decision zones.

DBM is, to our knowledge, the ﬁrst technique that

succeeds in visually depicting such classiﬁer decision

zones for any classiﬁer. However, DBM has several

limitations – it is slow, sensitive to parameter settings,

and produces noisy visualizations from which it is hard

to understand which are the shapes, topologies, and

extents of the decision zones.

In this paper, we propose Supervised Decision

Boundary Maps (SDBM), a supervised technique,

which improves DBM in four key directions:

Quality (C1):

SDBM produces decision maps that

allow for a clearer, and far less noise-prone, visual

separation of a higher number of decision zones from

real-world, complex, datasets, than DBM;

Scalability (C2):

SDBM is GPU accelerated and has

a complexity linear in the number of samples and di-

mensions, allowing the creation of megapixel maps in

Oliveira, A., Espadoto, M., Hirata Jr., R. and Telea, A.

SDBM: Supervised Decision Boundary Maps for Machine Learning Classiﬁers.

DOI: 10.5220/0010896200003124

In Proceedings of the 17th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2022) - Volume 3: IVAPP, pages 77-87

ISBN: 978-989-758-555-5; ISSN: 2184-4321

a few seconds on commodity hardware, in contrast to

the minutes needed by DBM;

Ease of Use (C3):

SDBM produces good results with

minimal or no parameter tuning;

Genericity (C4):

SDBM can construct decision

boundaries for any single-value classiﬁer.

We structure this paper as follows: Section 2 dis-

cusses related work on classiﬁer visualization. Sec-

tion 3 details our SDBM method. Section 4 presents

the results that support our contributions outlined

above. Section 5 discusses our method. Finally, Sec-

tion 6 concludes the paper.

2 BACKGROUND

We next introduce the notation used in this paper. Let

x = (x

, . . . , x

)

∈ R, 1 ≤ i ≤ n

be an

-dimensional

(

D) real-valued, labeled, observation, and let

D =

}

1 ≤ j ≤ N

be a dataset of

such samples, e.g.,

a table with

rows (samples) and

columns (dimen-

sions). Let

C = {c

}

1 ≤ k ≤ K

be the set of

class

labels used in

. Let

y = {y

∈ C}

1 ≤ j ≤ N

the class labels associated with each sample x

A classiﬁer is a function

f : R

→ C, (1)

that maps between data samples and class labels. The

classiﬁer

is typically obtained by using a training

algorithm over the dataset

. Common ML algorithms

are Logistic Regression (Cox, 1958), SVM (Cortes and

Vapnik, 1995), Random Forests (Breiman, 2001), and

Neural Networks, to name a few.

A Dimensionality Reduction (DR), or projection,

technique is a function

P : R

→ R

, (2)

where

q  n

, and typically

q = 2

. The projection

P(x)

of a sample

x ∈ R

is a

D point

p ∈ R

. Projecting a

set

yields thus a

D scatterplot, which we denote

next as

P(D)

. The inverse of

, denoted

−1

(p)

, maps

a qD point p to the high-dimensional space R

Decision Boundary Maps:

Given a classiﬁer

, a De-

cision Boundary Map (DBM) is a 2D image that shows

a representation of how

partitions the

data space

into decision zones. A decision zone is a set of 2D

points

for which

f (P

−1

(p)) = {c

∈ C}

– that is,

map high-dimensional points which are classiﬁed by

to the same class

. Class labels

are color-coded

in the decision maps. Decision zones are separated by

decision boundaries, which are pixels

whose labels

(colors) differ from those of at least one 8-neighbor

pixel in the DBM. The DBM shows, among other

things, how the high-dimensional space is effectively

partitioned by

into decision zones, how large these

zones are, how they are adjacent to each other, and

how smooth the decision boundaries between classes

are (Rodrigues et al., 2019). This gives insights on

whether the classiﬁer

has overﬁtted the training data,

and how well separated the data is, i.e., how difﬁcult

is the task of partitioning the high-dimensional space

to obtain good classiﬁcation accuracy. DBMs are a

step forward atop of the key observation in Rauber et

al. (Rauber et al., 2017b), which showed how multi-

dimensional projections aid deciding whether a high-

dimensional dataset is easily classiﬁable or not. Sim-

ply put, DBMs support the same task but provide more

information by ‘ﬁlling in’ the white gaps between the

points of a 2D scatterplot

P(D)

by extrapolating the

classiﬁer f .

The DBM technique, as introduced by Rodrigues

et al. (Rodrigues et al., 2019), relies heavily on direct

and inverse projections, to create the mappings

and

−1

. The direct mapping is used to create a 2D

scatterplot

P(D)

from the dataset

. The inverse

mapping

−1

creates synthetic

D data points from

all pixels

in the 2D bounding box of

P(D)

. These

points

−1

(p)

are then classiﬁed by

, and colored

by the assigned class labels

f (P

−1

(p))

. While this

approach is conceptually sound, it has two main

issues: (1) The inverse projection technique

−1

used,

iLAMP (Amorim et al., 2012), scales poorly to the

hundreds of thousands of points a dense pixel map

has. This was addressed in (Rodrigues et al., 2019)

by subsampling the 2D projection space into cells

larger than one pixel, sampling a few 2D pixels from

each cell, and next deciding the label (and thus color)

of each cell by majority voting on the classiﬁcation

of the inverse-projections of these samples. This

subsampling creates artifacts which are visible in the

highly jagged boundaries of the decision zones. (2)

Since the direct projections

used are unsupervised,

outliers in the data

can generate ‘islands’ of pixels

having a different label (and thus color) than their

neighbors. This creates spurious decision zones and

decision boundaries which next make the resulting

DBMs hard to analyze by the user, in particular when

the problem has several classes.

Dimensionality Reduction:

Both the original DBM

technique and our improved version SDBM rely heav-

ily on Dimensionality Reduction (DR) techniques.

Many DR techniques have been proposed over the

years, as reviewed in various surveys (Hoffman and

Grinstein, 2002; Maaten and Postma, 2009; Engel

et al., 2012; Sorzano et al., 2014; Liu et al., 2015;

IVAPP 2022 - 13th International Conference on Information Visualization Theory and Applications

Cunningham and Ghahramani, 2015; Xie et al., 2017;

Nonato and Aupetit, 2018; Espadoto et al., 2019a).

Below we describe a few representative ones, referring

to the aforementioned surveys for a more thorough

discussion.

Principal Component Analysis (Jolliffe, 1986)

(PCA) is one of the most popular DR techniques for

many decades, being easy to use, easy to interpret, and

scalable. However, PCA does not perform well for

data of high intrinsic dimensionality, and is thus not

the best option for data visualization tasks.

The Manifold Learning family of methods con-

tains techniques such as MDS (Torgerson, 1958),

Isomap (Tenenbaum et al., 2000), and LLE (Roweis

and Saul, 2000), which aim to capture nonlinear data

structure by mapping the high-dimensional manifold

on which data is located to 2D. These methods gen-

erally yield better results than PCA for visualization

tasks, but do not scale well computationally, and also

yield poor results when the intrinsic data dimensional-

ity is higher than two.

The SNE (Stochastic Neighborhood Embedding)

family of methods, of which the most popular member

is t-SNE (Maaten and Hinton, 2008), are very good for

visual tasks due to the visual cluster segregation they

produce. Yet, they can be hard to tune (Wattenberg,

2016), and typically have no out-of-sample capabil-

ity. Several reﬁnements of t-SNE improve speed, such

as tree-accelerated t-SNE (Maaten, 2014), hierarchi-

cal SNE (Pezzotti et al., 2016), and approximated t-

SNE (Pezzotti et al., 2017), and various GPU accel-

erations of t-SNE (Pezzotti et al., 2020; Chan et al.,

2018). Uniform Manifold Approximation and Pro-

jection (UMAP) (McInnes and Healy, 2018), while

not part of the SNE family, generates projections with

comparable quality to t-SNE, but much faster, and with

out-of-sample capability.

All above projection techniques work in an unsu-

pervised fashion, by using information on distances be-

tween data points in

to compute the projection

P(D)

Recently, (Espadoto et al., 2020) proposed Neural Net-

work Projection (NNP) to learn the projection

P(D)

computed by any user-selected technique

, from a

small subset

⊂ D

, using a deep learning regres-

sor. While slightly less accurate than the original

this technique is computationally linear in the size

and dimensionality of

, has out-of-sample capability,

is stable, and it simple to implement and parameter-

free. The same idea was used by NNInv (Espadoto

et al., 2019b) to learn the inverse mapping

−1

. These

approaches were next extended by Self-Supervised

Network Projection (SSNP) (Espadoto et al., 2021),

which can be used either in a self-supervised fashion,

by computing pseudo-labels by a generic clustering

algorithm on

, or in a supervised fashion (similar

to NNP), using ground-truth labels

coming with

SSNP’s supervised mode is key to the creation of our

proposed SDBM for the following reasons:

•

SSNP provides good cluster separation by parti-

tioning the data space

as a classiﬁer would do,

which is closely related to the original goal of

DBM;

•

SSNP provides both the direct and inverse map-

pings (

and

−1

) needed by DBM to generate

synthetic data points;

•

SSNP is GPU-accelerated, which makes SDBM

one to two magnitude orders faster than DBM.

3 METHOD

We next describe our proposed SDBM technique and

how it is different from its predecessor, DBM (see also

Fig. 1 for step-by step details of the SDBM pipeline):

LR, RF, etc

2. Create Mappings

2D → nD

1. Train classifier

3. Create grid

4. Create synthetic

nD points

5. Color pixels

HueEvaluate

SSNP

Figure 1: SDBM pipeline.

1. Train Classiﬁer:

Train the classiﬁer

to be visual-

ized using the dataset

and its class labels

. This step

is identical to DBM. Any single-class-output classiﬁer

f : R

→ C

can be used generically, e.g., Logistic Re-

gression (LR), Random Forests (RF), Support Vector

Machines (SVM), or neural networks.

2. Create Mappings:

Train SSNP to create the direct

and inverse projections

and

−1

based on

and

. This step is fundamentally different from DBM

which accepts any user-selected projection

and then

constructs

−1

by deep learning the 2D to

D mapping

using deep learning (Espadoto et al., 2019b) (see also

Sec. 2). This asymmetric design of DBM makes

−1

signiﬁcantly differ from the mathematical inverse of

for several points

, i.e.,

−1

(P(x)) 6= x

, which is

visible as jagged decision boundaries and noise-like

small islands scattered all over the dense maps (see

Fig. 5 later on). As we shall see in Sec. 4, the joint

computation of

and

−1

used by SDBM signiﬁcantly

reduces such artifacts.

3. Create 2D Grid:

Create an image

G ⊂ R

. This is

different from DBM which uses subsampling of the

2D projection space (see Sec. 2). In detail, SDBM

SDBM: Supervised Decision Boundary Maps for Machine Learning Classiﬁers

Logistic Regression SVM Random Forests Neural Network

10-class/700 dims 10-class/100 dims 2-class/700 dims 2-class/100 dims

Figure 2: Decision Boundary Maps (DBMs) created with SDBM for several classiﬁers (columns) and synthetic datasets (rows).

Lighter pixels represent training samples from the datasets D.

uses the full resolution of

to compute

P(D)

, but then

evaluates

−1

on a subsampled version thereof. In our

case, both

and

−1

use the full resolution image

For the experiments in this paper, we set the resolution

of G to 300

pixels.

4. Create Synthetic Data Points:

Use the trained

−1

to map each pixel

p ∈ G

to a high-dimensional

data point

x ∈ R

. This is similar to DBM, except the

use of a dense pixel grid and jointly-trained

and

−1

(see above).

5. Color Pixels:

Color all pixels

p ∈ G

by the values

f (P

−1

(p))

, i.e., the inferred classes of their corre-

sponding (synthetic) data points, using a categorical

color map. In this paper we use the ‘tab20’ color

map (Hunter, 2007). This is the same as DBM.

6. Encode Classiﬁer Conﬁdence (Optional):

For

classiﬁers

that provide the probability of a sample

belonging to a class

, we encode that probability

in the brightness of the pixel

that back-projects to

The lower the conﬁdence of the classiﬁer is, the darker

the pixel appears in the map, thereby informing the

user of the conﬁdence of the decision zone in that area.

This is the same as DBM.

4 RESULTS

We next present the results that support our claims

regarding SDBM. First, we show how our method

performs with synthetic data, where a perfect class

separation is possible by most classiﬁers (Sec. 4.1).

This allows us to verify how the technique performs

under a controlled setting where we know the ‘ground

truth’ shapes of the decision zones. Next, we show how

SDBM performs on more complex real-world datasets

and additional classiﬁers (Sec. 4.2) and also how it

compares with DBM. This supports our claim that our

technique can be generically used and that it improves

quality vs DBM. We next show how SDBM compares

to the original DBM speed-wise, thereby supporting

our claims of improved scalability (Sec. 4.3). Finally,

we provide full implementation details for SDBM

(Sec. 4.4).

IVAPP 2022 - 13th International Conference on Information Visualization Theory and Applications

Logistic Regression SVM Random Forests Neural Network

Reuters MNIST HAR FashionMNIST

0.828 0.848 0.849 0.849

0.981 0.972 0.982 0.988

0.889 0.950 0.940 0.943

0.893 0.887 0.874 0.882

Figure 3: Decision Boundary Maps (DBMs) created with SDBM for several classiﬁers (columns) and real-world datasets

(rows). Numbers inside each map indicate test accuracy obtained by each classiﬁer, bold indicating top performers. Lighter

pixels represent training samples from the datasets D.

4.1 Quality on Synthetic Datasets

We assess how SDBM performs in a controlled situ-

ation where the ground truth is known, i.e., datasets

with clear class separation and known shapes of the

expected decision zones. The datasets contain syn-

thetic Gaussian blobs with 5000 samples, with varied

dimensionality (100 and 700), and varied number of

classes (2 and 10). We used four different classiﬁers,

namely Logistic Regression, SVM (with a RBF ker-

nel), Random Forests (200 estimators), and a Neural

Network (multi-layer perceptron having 3 layers of

200 units).

Figure 2 shows the maps created using SDBM for

all the different classiﬁer and dataset combinations.

Decision zones are categorically colored. Projected

samples in

P(D)

are drawn colored also by their class,

but slightly brighter, so as to distinguish them from

the maps. We see that the decision zones are com-

pact and with smooth boundaries, as expected for such

simple classiﬁcation problems. They enclose the Gaus-

sian blobs with the same respective labels – e.g., the

red and blue zones for the 2-class, 100-dimensional

dataset in Fig. 2, top row, contain two clusters of light

red, respectively light blue, projected points. We also

see that the maps for Logistic Regression show almost

perfectly straight boundaries, which is a known fact for

this classiﬁer. In contrast, the more sophisticated clas-

siﬁers, such as Random Forests and Neural Networks,

create boundaries that are slightly more complex than

the others for the most complex dataset (Fig. 2, bottom

row, at the center of the maps for those classiﬁers).

4.2 Quality on Real-world Datasets

We next show how SDBM performs on real-world

datasets. These datasets are selected from publicly

available sources, matching the criteria of being high-

dimensional, reasonably large (thousands of samples),

and having a non-trivial data structure. They are also

frequently used in ML classiﬁcation evaluations and

projection evaluations.

SDBM: Supervised Decision Boundary Maps for Machine Learning Classiﬁers

Logistic Regression SVM Random Forests Neural Network

Reuters (w/ conf) Reuters HAR (w/ conf) HAR

Figure 4: Decision Boundary Maps created with SDBM for several classiﬁers, HAR and Reuters datasets. Columns show

different classiﬁers. Rows show different datasets, with and without conﬁdence encoded into brightness.

FashionMNIST

(Xiao et al., 2017)

70K samples of

K = 10

types of pieces of clothing, rendered as 28x28-

pixel gray scale images, ﬂattened to 784-element vec-

tors. We also use a subset of this dataset containing

only two classes, namely Ankle Boot and T-Shirt, to

provide an example of a problem where classes are

more easily separable. This dataset was downsampled

to 10K observations for all uses in this paper.

Human Activity Recognition (HAR)

(Anguita et al.,

2012)

10299 samples from 30 subjects performing

K = 6

activities of daily living used for human activ-

ity recognition, described with 561 dimensions that

encode 3-axial linear acceleration and 3-axial angular

velocity measured on the subjects.

MNIST

(LeCun and Cortes, 2010)

70K samples

K = 10

handwritten digits from 0 to 9, rendered

as 28x28-pixel gray scale images, ﬂattened to 784-

element vectors. This dataset was downsampled to

10K observations for all uses in this paper.

Reuters Newswire Dataset

(Thoma, 2017)

8432 ob-

servations of news report documents, from which 5000

attributes were extracted using TF-IDF (Salton and

McGill, 1986), a standard method in text processing.

This is a subset of the full dataset which contains data

for the K = 6 most frequent classes.

Figure 3 shows the maps created by SDBM for

these datasets, with the same types of classiﬁers used

in Sec. 4.1. Even though the current real-world

datasets are considerably more complex and harder

to separate into classes, the classiﬁers’ decision bound-

aries are clearly visible. Simpler classiﬁers (Logistic

Regression and SVM) show decision zones that are

more contiguous and have smoother, simpler, bound-

aries. More complex classiﬁers (Random Forests and

Neural Networks) show more complex shapes and

topologies of the decision zones. In particular, the

maps created for the Random Forest classiﬁers show

very jagged boundaries. This can be a result of having

an ensemble of classiﬁers working together.

Encoding Classiﬁer Conﬁdence:

Figure 4 shows

maps created by SDBM with classiﬁer conﬁdence en-

coded as brightness, as described in Sec. 3. This allows

us to see how different classiﬁers model probability

very differently, and thus produce different results.

IVAPP 2022 - 13th International Conference on Information Visualization Theory and Applications

The added value of encoding conﬁdence can be seen

if we compare the ﬁrst-vs-second, respectively third-

vs-fourth, rows in Fig. 4. The conﬁdence-encoding

maps show a smooth brightness gradient, dark close

to the decision boundaries (where colors change in

the images) and bright deep in the decision zones.

The effect is slightly reminiscent of shaded cushion

maps (van Wijk and van de Wetering, 1999), i.e., it

enhances the visual separation of the color-coded de-

cision zones. More importantly, the shading gradi-

ent effectively shows how conﬁdence increases as we

go deeper into the decision zones for different classi-

ﬁers: For example, for the HAR dataset, these shaded

bands are quite thin for Logistic Regression and SVM,

thicker and less informative for Random Forests, and

extremely and uniformly thin for Neural Networks.

This tells us that Neural Networks have an overall very

high conﬁdence everywhere (except very close to the

decision boundaries); Logistic Regression and SVM

are less conﬁdent close to the boundaries; and Random

Forests have a higher variation of conﬁdence over the

data space. For Random Forests, we see that the dark-

est region falls in the area of the central blue decision

zone and the top-right of the left yellow zone. This

are precisely the areas where the map of this classiﬁer

signiﬁcantly differs from those of all the other three

classiﬁers. Hence, we can infer that the isolated blue

decision zone that Random Forests created is likely

wrong, as it is low conﬁdence and different from what

all the other three classiﬁers created in that area. For

the Reuters dataset (Fig. 4 bottom row), we see that

all classiﬁers produced a beige region at the top left

corner. The conﬁdence information (brightness) shows

us that all classiﬁers but one (SVM) treat this region

as a low conﬁdence one. This can be explained by the

total absence of training samples in that region. More

importantly, this tells us that the behavior of SVM in

this region is likely wrong.

Conﬁdence visualization also serves in quickly and

globally assessing the overall quality of a trained clas-

siﬁer. Consider e.g. the Reuters dataset (Fig. 4 bot-

tom row). Compared to all other three rows in Fig. 4,

the decision maps for this dataset are darker. This

shows that this dataset is harder to extrapolate from

during inference. Note that this is not the same as the

usual testing-after-training in ML. Indeed, for testing,

one needs to ‘reserve’ a set of samples unseen dur-

ing training to evaluate the trained classiﬁer on. In

contrast, SDBM’s decision maps do not need to do

this as they synthesize ‘testing’ samples on the ﬂy via

the inverse projection

−1

. Moreover, classical ML

testing only gives a global or per-class accuracy. In

contrast, SDBM gives a per-region-of-the-data-space

conﬁdence, encoded by brightness.

Comparison with the Original DBM:

Figure 5

shows maps created by SDBM side-by-side with maps

created by the original DBM technique, using Logistic

Regression, Random Forest and k-NN classiﬁers, for

three real-world datasets. In this experiment, we used

UMAP (McInnes and Healy, 2018) as the direct projec-

tion for DBM, and iLAMP (Amorim et al., 2012) for

the inverse projection, respectively Several important

observations can be made, as follows.

First, we see that the projections

P(D)

of the same

datasets are not the same with DBM and SDBM –

compare the bright-colored dots in the correspond-

ing ﬁgures. This is expected, since, as explained in

Sec. 3, DBM employs a user-chosen projection tech-

nique

, whereas SDBM learns

from the label-based

clustering of the data, following the SSNP method

(see Sec. 3). Since the projections

P(D)

of the same

datasets differ for the two methods, it is expected that

the overall shapes of the ensuing decision boundaries

will also differ – see e.g. the difference between the

nearly horizontal decision boundary between the blue

and red zones for Random Forests with DBM for Fash-

ionMNIST (2-class) and the angled boundary between

the same zones for the same classiﬁer, same dataset,

with SDBM (Fig. 5, middle row, two leftmost images).

For the relatively simple classiﬁcation problem that

FashionMNIST (2-class) is, this is not a problem. Both

DBM and SDBM produce useful and usable renditions

of the two resulting decision zones, showing that this

classiﬁcation problem succeeded with no issues.

When considering more difﬁcult datasets (Fashion-

MNIST 10-class or HAR), the situation is dramatically

different: DBM shows highly noisy pictures, where

it is even hard to say where and which are the actual

decision zones. These images suggest that none of

the three tested classiﬁers could correctly handle these

datasets, in the sense that they would change deci-

sions extremely rapidly and randomly as points only

slightly change over the data space. This is known not

to be the case for these datasets and classiﬁers. Logis-

tic Regression has built-in limitations of how quickly

its decision boundaries can change (Rodrigues et al.,

2019). k-NN is also known to construct essentially

a Voronoi diagram around the same-class samples in

the

D space, partitioning that space into cells whose

boundaries are smooth manifolds. DBM does not show

any such behavior (Fig. 5, third and ﬁfth columns). In

contrast, SDBM shows a far lower noise level and

far smoother, contiguous, decision zones and bound-

aries. Even though we do not have formal ground

truth on how the zones and boundaries of these dataset-

classiﬁer combinations actually look, SDBM matches

better the knowledge we have on these problems than

DBM.

SDBM: Supervised Decision Boundary Maps for Machine Learning Classiﬁers

DBM SDBM

kNN Random Forests Logistic Regression

FashionMNIST (2-class) FashionMNIST (10-class) HAR

DBM SDBM

Figure 5: Comparison between SDBM and DBM using three different datasets and three classiﬁers.

4.3 Computational Scalability

We next study the scalability of SDBM and compare

it to the original DBM method. For this, we created

maps using synthetic Gaussian blobs datasets with 5

clusters, varying the dimensionality from 10 to 500,

and varying the map size from

300

pixels.

We did not use larger maps since the speed-trends

were already clear from these sizes, with DBM getting

considerably slower than SDBM. Figure 6 shows the

running times of both methods as a function of both the

grid size (horizontal axis) and dataset dimensionality

(different-color lines). We see that DBM’s runtime

increases quickly with dimensionality, taking about 5

minutes to create a

300

map for the 500-dimensional

dataset.

In contrast, SDBM is over an order of magni-

tude faster, taking roughly 7 seconds to run for the

same dataset. Also, we see that SDBM’s speed only

marginally depends on the dimensionality, whereas

this is a major slowdown factor for DBM. With re-

spect to the number of samples, we see that both meth-

ods exhibit similar trends, with SDBM being closer

to a linear trend than DBM. However, the slope of the

SDBM graphs is smaller than for DBM for the same

dimensionality. All in all, this shows that SDBM is

signiﬁcantly more scalable than DBM. This can be ex-

plained by the fact that SSNP, which underlies SDBM,

jointly trains both the direct and inverse projections by

deep learning. As this is GPU-accelerated, linear in

the sample and dimension counts both for training and

inference, and does not need to use different resolu-

tions and sampling tricks for accelerating the 2D to

mapping (see Sec. 3). In contrast, DBM uses UMAP

and iLAMP for the direct, respectively, inverse projec-

tions (as mentioned earlier). None of these techniques

is GPU-accelerated.

4.4 Implementation Details

All experiments presented above were run on a dual 8-

core Intel Xeon Silver 4110 with 256 GB RAM and an

NVidia GeForce RTX 2070 GPU with 8 GB VRAM.

Table 1 lists all open-source software libraries used

to build SDBM and the other tested techniques. Our

implementation, plus all code used in this experiment,

are publicly available at (The Authors, 2021).

Table 1: Software packages used in the evaluation.

Technique Software used publicly available at

SSNP keras.io (TensorFlow backend) (Chollet and others, 2015)

UMAP github.com/lmcinnes/umap (McInnes and Healy, 2018)

5 DISCUSSION

We discuss how our technique performs with respect

to the criteria laid out in Section 1.

Quality (C1):

SDBM is able to create maps that show

classiﬁer decision boundaries very clearly, and, most

importantly, much clearer than the maps created with

IVAPP 2022 - 13th International Conference on Information Visualization Theory and Applications

Figure 6: Plot showing the order of growth of time used to create maps of increasing size using DBM and SDBM, using

synthetic datasets of varying dimensionality. Vertical axis is in logarithmic scale.

the original DBM. For the same dataset-classiﬁer com-

binations, SDBM’s maps show signiﬁcantly less noise,

more compact decision zones, and smoother decision

boundaries, than DBM. These results are in line with

what we expect for dataset-classiﬁer combinations for

which we have ground-truth knowledge about their

decision zones and boundaries (see Fig. 5 and related

text). As such, we conclude that SDBM captures the

actual decision zones better than DBM can do.

Scalability (C2):

SDBM is an order of magnitude

faster than DBM. Since SDBM scales linearly in the

number of observations during inference/drawing, and

it is end-to-end GPU-accelerated, it is able to generate

maps having hundreds of thousands of pixels in a few

seconds, which makes it practical for handling large

datasets and rendering highly detailed decision maps.

Ease of use (C3): SDBM produces good results with

minimal tuning. The single performance-sensitive set-

ting is the size of the map image. All maps in this paper

have

300

pixels. As the ﬁgures show, this resolution

is already sufﬁcient for rendering detailed decision

maps for all the tested dataset-classiﬁer combination.

Compared to DBM, SDBM tuning is far simpler, as

it does not require tuning of cell and sample sizes re-

quired by the former (for details of DBM tuning, we

refer to (Rodrigues et al., 2019)).

Genericity (C4):

As for the original DBM method,

SDBM is agnostic to the nature and dimensionality of

the input data, and to the classiﬁer being visualized.

We show that SDBM achieves high quality on datasets

of different natures and coming from a wide range

of application domains, and with classiﬁers based on

quite different algorithms. As such, SDBM does not

trade any ﬂexibility that DBM already offered, but

increases quality, scalability, and ease of use, as ex-

plained above.

Limitations:

SDBM shares a few limitations with

DBM. First and foremost, it is hard to formally assess

the quality of the decision maps it produces for dataset-

classiﬁer combinations for which we do not have clear

ground-truth on the shape and position of their deci-

sion zones and boundaries. Current testing shown in

this paper has outlined that SDBM produces results

fully in line with known ground truth for such simple

situations. However, this does not formally guarantee

that the same is true for more complex datasets and

any classiﬁers. Finding ways to assess this is an open

problem to be studied in future work. Secondly, the

interpretation of the SDBM maps can be enhanced.

Examples shown in this paper outlined how such maps

can help ﬁnding out whether a trained classiﬁer can

generalize well, and how far, from its training set, and

how different classiﬁer-dataset combinations can be

compared by such maps. Yet, such evidence is qualita-

tive. A more formal study showing how users actually

interpret such maps to extract quantitative information

on the visualized classiﬁcation problems is needed.

6 CONCLUSION

We have presented SDBM, a new method for produc-

ing classiﬁer Decision Boundary Maps. Compared to

the only similar technique we are aware of – DBM –

our method presents several desirable characteristics.

First and foremost, it is able to create decision maps

which are far smoother and less noisy than those cre-

ated by DBM and also match the known ground-truth

of the visualized classiﬁcation problems far better than

DBM, therefore allowing users to interpret the studied

classiﬁers with less confusion. Secondly, SDBM is

about an order of magnitude faster than DBM due to

its joint computation of direct and inverse projections

SDBM: Supervised Decision Boundary Maps for Machine Learning Classiﬁers

on a ﬁxed-resolution image. Finally, SDBM has virtu-

ally no parameters to tune (apart from the resolution of

the desired ﬁnal image) which makes it easier to use

than DBM.

Future work can target several directions. We be-

lieve a very relevant one to be the generation of maps

for multi-output classiﬁers, i.e., classiﬁers that can out-

put more than a single class for a sample. Secondly,

we consider organizing more quantitative studies to

actually gauge which are the interpretation errors that

SDBM maps generate when users consider them to

assess and/or compare the behavior of different clas-

siﬁers, which is the core use-case that decision maps

have been proposed for. Thirdly, we consider adapting

SDBM to help the understanding of semantic segmen-

tation models. Last but not least, the packaging of

SDBM into a reusable library that can be integrated

into typical ML pipelines can help it gain widespread

usage.

ACKNOWLEDGMENTS

This study was ﬁnanced in part by the Coordena

c¸

de Aperfei

c¸

oamento de Pessoal de N

ıvel Superior -

Brasil (CAPES) - Finance Code 001, and by FAPESP

grants 2015/22308-2, 2017/25835-9 and 2020/13275-

1, Brazil.

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