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<section id="computational-performance">
<span id="id1"></span><h1><span class="section-number">8.2. </span>Computational Performance<a class="headerlink" href="#computational-performance" title="Link to this heading">¶</a></h1>
<p>For some applications the performance (mainly latency and throughput at
prediction time) of estimators is crucial. It may also be of interest to
consider the training throughput but this is often less important in a
production setup (where it often takes place offline).</p>
<p>We will review here the orders of magnitude you can expect from a number of
scikit-learn estimators in different contexts and provide some tips and
tricks for overcoming performance bottlenecks.</p>
<p>Prediction latency is measured as the elapsed time necessary to make a
prediction (e.g. in micro-seconds). Latency is often viewed as a distribution
and operations engineers often focus on the latency at a given percentile of
this distribution (e.g. the 90 percentile).</p>
<p>Prediction throughput is defined as the number of predictions the software can
deliver in a given amount of time (e.g. in predictions per second).</p>
<p>An important aspect of performance optimization is also that it can hurt
prediction accuracy. Indeed, simpler models (e.g. linear instead of
non-linear, or with fewer parameters) often run faster but are not always able
to take into account the same exact properties of the data as more complex ones.</p>
<section id="prediction-latency">
<h2><span class="section-number">8.2.1. </span>Prediction Latency<a class="headerlink" href="#prediction-latency" title="Link to this heading">¶</a></h2>
<p>One of the most straight-forward concerns one may have when using/choosing a
machine learning toolkit is the latency at which predictions can be made in a
production environment.</p>
<dl class="simple">
<dt>The main factors that influence the prediction latency are</dt><dd><ol class="arabic simple">
<li><p>Number of features</p></li>
<li><p>Input data representation and sparsity</p></li>
<li><p>Model complexity</p></li>
<li><p>Feature extraction</p></li>
</ol>
</dd>
</dl>
<p>A last major parameter is also the possibility to do predictions in bulk or
one-at-a-time mode.</p>
<section id="bulk-versus-atomic-mode">
<h3><span class="section-number">8.2.1.1. </span>Bulk versus Atomic mode<a class="headerlink" href="#bulk-versus-atomic-mode" title="Link to this heading">¶</a></h3>
<p>In general doing predictions in bulk (many instances at the same time) is
more efficient for a number of reasons (branching predictability, CPU cache,
linear algebra libraries optimizations etc.). Here we see on a setting
with few features that independently of estimator choice the bulk mode is
always faster, and for some of them by 1 to 2 orders of magnitude:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_prediction_latency.html"><img alt="atomic_prediction_latency" src="../_images/sphx_glr_plot_prediction_latency_001.png" style="width: 800.0px; height: 480.0px;" /></a></strong></p><p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_prediction_latency.html"><img alt="bulk_prediction_latency" src="../_images/sphx_glr_plot_prediction_latency_002.png" style="width: 800.0px; height: 480.0px;" /></a></strong></p><p>To benchmark different estimators for your case you can simply change the
<code class="docutils literal notranslate"><span class="pre">n_features</span></code> parameter in this example:
<a class="reference internal" href="../auto_examples/applications/plot_prediction_latency.html#sphx-glr-auto-examples-applications-plot-prediction-latency-py"><span class="std std-ref">Prediction Latency</span></a>. This should give
you an estimate of the order of magnitude of the prediction latency.</p>
</section>
<section id="configuring-scikit-learn-for-reduced-validation-overhead">
<h3><span class="section-number">8.2.1.2. </span>Configuring Scikit-learn for reduced validation overhead<a class="headerlink" href="#configuring-scikit-learn-for-reduced-validation-overhead" title="Link to this heading">¶</a></h3>
<p>Scikit-learn does some validation on data that increases the overhead per
call to <code class="docutils literal notranslate"><span class="pre">predict</span></code> and similar functions. In particular, checking that
features are finite (not NaN or infinite) involves a full pass over the
data. If you ensure that your data is acceptable, you may suppress
checking for finiteness by setting the environment variable
<code class="docutils literal notranslate"><span class="pre">SKLEARN_ASSUME_FINITE</span></code> to a non-empty string before importing
scikit-learn, or configure it in Python with <a class="reference internal" href="../modules/generated/sklearn.set_config.html#sklearn.set_config" title="sklearn.set_config"><code class="xref py py-func docutils literal notranslate"><span class="pre">set_config</span></code></a>.
For more control than these global settings, a <a class="reference internal" href="../modules/generated/sklearn.config_context.html#sklearn.config_context" title="sklearn.config_context"><code class="xref py py-func docutils literal notranslate"><span class="pre">config_context</span></code></a>
allows you to set this configuration within a specified context:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">sklearn</span>
<span class="gp">>>> </span><span class="k">with</span> <span class="n">sklearn</span><span class="o">.</span><span class="n">config_context</span><span class="p">(</span><span class="n">assume_finite</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">pass</span> <span class="c1"># do learning/prediction here with reduced validation</span>
</pre></div>
</div>
<p>Note that this will affect all uses of
<a class="reference internal" href="../modules/generated/sklearn.utils.assert_all_finite.html#sklearn.utils.assert_all_finite" title="sklearn.utils.assert_all_finite"><code class="xref py py-func docutils literal notranslate"><span class="pre">assert_all_finite</span></code></a> within the context.</p>
</section>
<section id="influence-of-the-number-of-features">
<h3><span class="section-number">8.2.1.3. </span>Influence of the Number of Features<a class="headerlink" href="#influence-of-the-number-of-features" title="Link to this heading">¶</a></h3>
<p>Obviously when the number of features increases so does the memory
consumption of each example. Indeed, for a matrix of <span class="math notranslate nohighlight">\(M\)</span> instances
with <span class="math notranslate nohighlight">\(N\)</span> features, the space complexity is in <span class="math notranslate nohighlight">\(O(NM)\)</span>.
From a computing perspective it also means that the number of basic operations
(e.g., multiplications for vector-matrix products in linear models) increases
too. Here is a graph of the evolution of the prediction latency with the
number of features:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_prediction_latency.html"><img alt="influence_of_n_features_on_latency" src="../_images/sphx_glr_plot_prediction_latency_003.png" style="width: 800.0px; height: 480.0px;" /></a></strong></p><p>Overall you can expect the prediction time to increase at least linearly with
the number of features (non-linear cases can happen depending on the global
memory footprint and estimator).</p>
</section>
<section id="influence-of-the-input-data-representation">
<h3><span class="section-number">8.2.1.4. </span>Influence of the Input Data Representation<a class="headerlink" href="#influence-of-the-input-data-representation" title="Link to this heading">¶</a></h3>
<p>Scipy provides sparse matrix data structures which are optimized for storing
sparse data. The main feature of sparse formats is that you don’t store zeros
so if your data is sparse then you use much less memory. A non-zero value in
a sparse (<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/sparse.html">CSR or CSC</a>)
representation will only take on average one 32bit integer position + the 64
bit floating point value + an additional 32bit per row or column in the matrix.
Using sparse input on a dense (or sparse) linear model can speedup prediction
by quite a bit as only the non zero valued features impact the dot product
and thus the model predictions. Hence if you have 100 non zeros in 1e6
dimensional space, you only need 100 multiply and add operation instead of 1e6.</p>
<p>Calculation over a dense representation, however, may leverage highly optimized
vector operations and multithreading in BLAS, and tends to result in fewer CPU
cache misses. So the sparsity should typically be quite high (10% non-zeros
max, to be checked depending on the hardware) for the sparse input
representation to be faster than the dense input representation on a machine
with many CPUs and an optimized BLAS implementation.</p>
<p>Here is sample code to test the sparsity of your input:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">sparsity_ratio</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">count_nonzero</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">/</span> <span class="nb">float</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"input sparsity ratio:"</span><span class="p">,</span> <span class="n">sparsity_ratio</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
</pre></div>
</div>
<p>As a rule of thumb you can consider that if the sparsity ratio is greater
than 90% you can probably benefit from sparse formats. Check Scipy’s sparse
matrix formats <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/sparse.html">documentation</a>
for more information on how to build (or convert your data to) sparse matrix
formats. Most of the time the <code class="docutils literal notranslate"><span class="pre">CSR</span></code> and <code class="docutils literal notranslate"><span class="pre">CSC</span></code> formats work best.</p>
</section>
<section id="influence-of-the-model-complexity">
<h3><span class="section-number">8.2.1.5. </span>Influence of the Model Complexity<a class="headerlink" href="#influence-of-the-model-complexity" title="Link to this heading">¶</a></h3>
<p>Generally speaking, when model complexity increases, predictive power and
latency are supposed to increase. Increasing predictive power is usually
interesting, but for many applications we would better not increase
prediction latency too much. We will now review this idea for different
families of supervised models.</p>
<p>For <a class="reference internal" href="../modules/classes.html#module-sklearn.linear_model" title="sklearn.linear_model"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.linear_model</span></code></a> (e.g. Lasso, ElasticNet,
SGDClassifier/Regressor, Ridge & RidgeClassifier,
PassiveAggressiveClassifier/Regressor, LinearSVC, LogisticRegression…) the
decision function that is applied at prediction time is the same (a dot product)
, so latency should be equivalent.</p>
<p>Here is an example using
<a class="reference internal" href="../modules/generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> with the
<code class="docutils literal notranslate"><span class="pre">elasticnet</span></code> penalty. The regularization strength is globally controlled by
the <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter. With a sufficiently high <code class="docutils literal notranslate"><span class="pre">alpha</span></code>,
one can then increase the <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> parameter of <code class="docutils literal notranslate"><span class="pre">elasticnet</span></code> to
enforce various levels of sparsity in the model coefficients. Higher sparsity
here is interpreted as less model complexity as we need fewer coefficients to
describe it fully. Of course sparsity influences in turn the prediction time
as the sparse dot-product takes time roughly proportional to the number of
non-zero coefficients.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_model_complexity_influence.html"><img alt="en_model_complexity" src="../_images/sphx_glr_plot_model_complexity_influence_001.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>For the <a class="reference internal" href="../modules/classes.html#module-sklearn.svm" title="sklearn.svm"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.svm</span></code></a> family of algorithms with a non-linear kernel,
the latency is tied to the number of support vectors (the fewer the faster).
Latency and throughput should (asymptotically) grow linearly with the number
of support vectors in a SVC or SVR model. The kernel will also influence the
latency as it is used to compute the projection of the input vector once per
support vector. In the following graph the <code class="docutils literal notranslate"><span class="pre">nu</span></code> parameter of
<a class="reference internal" href="../modules/generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a> was used to influence the number of
support vectors.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_model_complexity_influence.html"><img alt="nusvr_model_complexity" src="../_images/sphx_glr_plot_model_complexity_influence_002.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>For <a class="reference internal" href="../modules/classes.html#module-sklearn.ensemble" title="sklearn.ensemble"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.ensemble</span></code></a> of trees (e.g. RandomForest, GBT,
ExtraTrees, etc.) the number of trees and their depth play the most
important role. Latency and throughput should scale linearly with the number
of trees. In this case we used directly the <code class="docutils literal notranslate"><span class="pre">n_estimators</span></code> parameter of
<a class="reference internal" href="../modules/generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_model_complexity_influence.html"><img alt="gbt_model_complexity" src="../_images/sphx_glr_plot_model_complexity_influence_003.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>In any case be warned that decreasing model complexity can hurt accuracy as
mentioned above. For instance a non-linearly separable problem can be handled
with a speedy linear model but prediction power will very likely suffer in
the process.</p>
</section>
<section id="feature-extraction-latency">
<h3><span class="section-number">8.2.1.6. </span>Feature Extraction Latency<a class="headerlink" href="#feature-extraction-latency" title="Link to this heading">¶</a></h3>
<p>Most scikit-learn models are usually pretty fast as they are implemented
either with compiled Cython extensions or optimized computing libraries.
On the other hand, in many real world applications the feature extraction
process (i.e. turning raw data like database rows or network packets into
numpy arrays) governs the overall prediction time. For example on the Reuters
text classification task the whole preparation (reading and parsing SGML
files, tokenizing the text and hashing it into a common vector space) is
taking 100 to 500 times more time than the actual prediction code, depending on
the chosen model.</p>
<blockquote>
<div></div></blockquote>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_out_of_core_classification.html"><img alt="prediction_time" src="../_images/sphx_glr_plot_out_of_core_classification_004.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>In many cases it is thus recommended to carefully time and profile your
feature extraction code as it may be a good place to start optimizing when
your overall latency is too slow for your application.</p>
</section>
</section>
<section id="prediction-throughput">
<h2><span class="section-number">8.2.2. </span>Prediction Throughput<a class="headerlink" href="#prediction-throughput" title="Link to this heading">¶</a></h2>
<p>Another important metric to care about when sizing production systems is the
throughput i.e. the number of predictions you can make in a given amount of
time. Here is a benchmark from the
<a class="reference internal" href="../auto_examples/applications/plot_prediction_latency.html#sphx-glr-auto-examples-applications-plot-prediction-latency-py"><span class="std std-ref">Prediction Latency</span></a> example that measures
this quantity for a number of estimators on synthetic data:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/applications/plot_prediction_latency.html"><img alt="throughput_benchmark" src="../_images/sphx_glr_plot_prediction_latency_004.png" style="width: 800.0px; height: 480.0px;" /></a></strong></p><p>These throughputs are achieved on a single process. An obvious way to
increase the throughput of your application is to spawn additional instances
(usually processes in Python because of the
<a class="reference external" href="https://wiki.python.org/moin/GlobalInterpreterLock">GIL</a>) that share the
same model. One might also add machines to spread the load. A detailed
explanation on how to achieve this is beyond the scope of this documentation
though.</p>
</section>
<section id="tips-and-tricks">
<h2><span class="section-number">8.2.3. </span>Tips and Tricks<a class="headerlink" href="#tips-and-tricks" title="Link to this heading">¶</a></h2>
<section id="linear-algebra-libraries">
<h3><span class="section-number">8.2.3.1. </span>Linear algebra libraries<a class="headerlink" href="#linear-algebra-libraries" title="Link to this heading">¶</a></h3>
<p>As scikit-learn relies heavily on Numpy/Scipy and linear algebra in general it
makes sense to take explicit care of the versions of these libraries.
Basically, you ought to make sure that Numpy is built using an optimized <a class="reference external" href="https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms">BLAS</a> /
<a class="reference external" href="https://en.wikipedia.org/wiki/LAPACK">LAPACK</a> library.</p>
<p>Not all models benefit from optimized BLAS and Lapack implementations. For
instance models based on (randomized) decision trees typically do not rely on
BLAS calls in their inner loops, nor do kernel SVMs (<code class="docutils literal notranslate"><span class="pre">SVC</span></code>, <code class="docutils literal notranslate"><span class="pre">SVR</span></code>,
<code class="docutils literal notranslate"><span class="pre">NuSVC</span></code>, <code class="docutils literal notranslate"><span class="pre">NuSVR</span></code>). On the other hand a linear model implemented with a
BLAS DGEMM call (via <code class="docutils literal notranslate"><span class="pre">numpy.dot</span></code>) will typically benefit hugely from a tuned
BLAS implementation and lead to orders of magnitude speedup over a
non-optimized BLAS.</p>
<p>You can display the BLAS / LAPACK implementation used by your NumPy / SciPy /
scikit-learn install with the following command:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">python</span> <span class="o">-</span><span class="n">c</span> <span class="s2">"import sklearn; sklearn.show_versions()"</span>
</pre></div>
</div>
<dl class="simple">
<dt>Optimized BLAS / LAPACK implementations include:</dt><dd><ul class="simple">
<li><p>Atlas (need hardware specific tuning by rebuilding on the target machine)</p></li>
<li><p>OpenBLAS</p></li>
<li><p>MKL</p></li>
<li><p>Apple Accelerate and vecLib frameworks (OSX only)</p></li>
</ul>
</dd>
</dl>
<p>More information can be found on the <a class="reference external" href="https://numpy.org/install/">NumPy install page</a>
and in this
<a class="reference external" href="https://danielnouri.org/notes/2012/12/19/libblas-and-liblapack-issues-and-speed,-with-scipy-and-ubuntu/">blog post</a>
from Daniel Nouri which has some nice step by step install instructions for
Debian / Ubuntu.</p>
</section>
<section id="limiting-working-memory">
<span id="working-memory"></span><h3><span class="section-number">8.2.3.2. </span>Limiting Working Memory<a class="headerlink" href="#limiting-working-memory" title="Link to this heading">¶</a></h3>
<p>Some calculations when implemented using standard numpy vectorized operations
involve using a large amount of temporary memory. This may potentially exhaust
system memory. Where computations can be performed in fixed-memory chunks, we
attempt to do so, and allow the user to hint at the maximum size of this
working memory (defaulting to 1GB) using <a class="reference internal" href="../modules/generated/sklearn.set_config.html#sklearn.set_config" title="sklearn.set_config"><code class="xref py py-func docutils literal notranslate"><span class="pre">set_config</span></code></a> or
<a class="reference internal" href="../modules/generated/sklearn.config_context.html#sklearn.config_context" title="sklearn.config_context"><code class="xref py py-func docutils literal notranslate"><span class="pre">config_context</span></code></a>. The following suggests to limit temporary working
memory to 128 MiB:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">sklearn</span>
<span class="gp">>>> </span><span class="k">with</span> <span class="n">sklearn</span><span class="o">.</span><span class="n">config_context</span><span class="p">(</span><span class="n">working_memory</span><span class="o">=</span><span class="mi">128</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">pass</span> <span class="c1"># do chunked work here</span>
</pre></div>
</div>
<p>An example of a chunked operation adhering to this setting is
<a class="reference internal" href="../modules/generated/sklearn.metrics.pairwise_distances_chunked.html#sklearn.metrics.pairwise_distances_chunked" title="sklearn.metrics.pairwise_distances_chunked"><code class="xref py py-func docutils literal notranslate"><span class="pre">pairwise_distances_chunked</span></code></a>, which facilitates computing
row-wise reductions of a pairwise distance matrix.</p>
</section>
<section id="model-compression">
<h3><span class="section-number">8.2.3.3. </span>Model Compression<a class="headerlink" href="#model-compression" title="Link to this heading">¶</a></h3>
<p>Model compression in scikit-learn only concerns linear models for the moment.
In this context it means that we want to control the model sparsity (i.e. the
number of non-zero coordinates in the model vectors). It is generally a good
idea to combine model sparsity with sparse input data representation.</p>
<p>Here is sample code that illustrates the use of the <code class="docutils literal notranslate"><span class="pre">sparsify()</span></code> method:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">clf</span> <span class="o">=</span> <span class="n">SGDRegressor</span><span class="p">(</span><span class="n">penalty</span><span class="o">=</span><span class="s1">'elasticnet'</span><span class="p">,</span> <span class="n">l1_ratio</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span><span class="o">.</span><span class="n">sparsify</span><span class="p">()</span>
<span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
</pre></div>
</div>
<p>In this example we prefer the <code class="docutils literal notranslate"><span class="pre">elasticnet</span></code> penalty as it is often a good
compromise between model compactness and prediction power. One can also
further tune the <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> parameter (in combination with the
regularization strength <code class="docutils literal notranslate"><span class="pre">alpha</span></code>) to control this tradeoff.</p>
<p>A typical <a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/main/benchmarks/bench_sparsify.py">benchmark</a>
on synthetic data yields a >30% decrease in latency when both the model and
input are sparse (with 0.000024 and 0.027400 non-zero coefficients ratio
respectively). Your mileage may vary depending on the sparsity and size of
your data and model.
Furthermore, sparsifying can be very useful to reduce the memory usage of
predictive models deployed on production servers.</p>
</section>
<section id="model-reshaping">
<h3><span class="section-number">8.2.3.4. </span>Model Reshaping<a class="headerlink" href="#model-reshaping" title="Link to this heading">¶</a></h3>
<p>Model reshaping consists in selecting only a portion of the available features
to fit a model. In other words, if a model discards features during the
learning phase we can then strip those from the input. This has several
benefits. Firstly it reduces memory (and therefore time) overhead of the
model itself. It also allows to discard explicit
feature selection components in a pipeline once we know which features to
keep from a previous run. Finally, it can help reduce processing time and I/O
usage upstream in the data access and feature extraction layers by not
collecting and building features that are discarded by the model. For instance
if the raw data come from a database, it can make it possible to write simpler
and faster queries or reduce I/O usage by making the queries return lighter
records.
At the moment, reshaping needs to be performed manually in scikit-learn.
In the case of sparse input (particularly in <code class="docutils literal notranslate"><span class="pre">CSR</span></code> format), it is generally
sufficient to not generate the relevant features, leaving their columns empty.</p>
</section>
<section id="links">
<h3><span class="section-number">8.2.3.5. </span>Links<a class="headerlink" href="#links" title="Link to this heading">¶</a></h3>
<blockquote>
<div><ul class="simple">
<li><p><a class="reference internal" href="../developers/performance.html#performance-howto"><span class="std std-ref">scikit-learn developer performance documentation</span></a></p></li>
<li><p><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/sparse.html">Scipy sparse matrix formats documentation</a></p></li>
</ul>
</div></blockquote>
</section>
</section>
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