Thanks @akahard2dj . I think the goal of the issue is not to put the example back, since it was removed and moved around in #10700, but to change and fix the links on the homepage. You can see how the examples have been moved on the original PR, and fix the links accordingly.

@adrinjalali Thanks for your valuable comment.
I will check #10700.

The example replacing examples/covariance/plot_outlier_detection.py is examples/plot_anomaly_comparison.py.

You need to change both line 256 in doc/conf.py

carousel_thumbs = {'sphx_glr_plot_classifier_comparison_001.png': 600,
                   'sphx_glr_plot_outlier_detection_003.png': 372,
                   'sphx_glr_plot_gpr_co2_001.png': 350,
                   'sphx_glr_plot_adaboost_twoclass_001.png': 372,
                   'sphx_glr_plot_compare_methods_001.png': 349}

and lines 152-153 in doc/themes/scikit-learn/layout.html

<div class="item">
  <a href="{{pathto('auto_examples/covariance/plot_outlier_detection') }}">
    <img src="_images/sphx_glr_plot_outlier_detection_003_carousel.png"></a>
</div>

003 in the filename should be replaced by 001.

@albertcthomas Really thanks. This commit is entirely based on @albertcthomas 's comment.
plot_outlier_detection example is changed to novelty_detection.
So I also fixed that previous thumbs image of sphx_glr_plot_outlier_detection_00x.png to sphx_glr_plot_lof_novelty_detection_001.png.

I upload a render image.
In my option, all requests is fixed.
Please review the final draft.

restarted Circle, will merge when green.