added new footnote

paper.bib

@@ -135,15 +135,17 @@ @article{gans
 }
 
 
-@misc{Julia,
-      title={Julia: A Fast Dynamic Language for Technical Computing}, 
-      author={Jeff Bezanson and Stefan Karpinski and Viral B. Shah and Alan Edelman},
-      year={2012},
-      eprint={1209.5145},
-      archivePrefix={arXiv},
-      primaryClass={cs.PL},
-      doi={10.48550/arXiv.1209.5145},
-      url={https://doi.org/10.48550/arXiv.1209.5145}
+@article{Julia,
+    title={Julia: A fresh approach to numerical computing},
+    author={Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B},
+    journal={SIAM {R}eview},
+    volume={59},
+    number={1},
+    pages={65--98},
+    year={2017},
+    publisher={SIAM},
+    doi={10.1137/141000671},
+    url={https://epubs.siam.org/doi/10.1137/141000671}
 }
 
 @BOOK{GLM,

paper.md

@@ -39,7 +39,9 @@ Principal component analysis (PCA) [@PCA1; @PCA2; @PCA3] is popular for compress
 
 EPCA is used in reinforcement learning [@Roy], sample debiasing [@debiasing], and compositional analysis [@gans]. Wider adoption, however, remains limited due to the lack of implementations. The only other EPCA package is written in MATLAB and supports just one distribution [@epca-MATLAB]. This is surprising, as other Bregman-based optimization techniques have been successful in areas like mass spectrometry [@spectrum], ultrasound denoising [@ultrasound], topological data analysis [@topological], and robust clustering [@clustering]. These successes suggest that EPCA holds untapped potential in signal processing and machine learning.
 
-The absence of a general EPCA library likely stems from the limited interoperability between fast symbolic differentiation and optimization libraries in popular languages like Python and C. Julia, by contrast, uses multiple dispatch which promotes high levels of generic code reuse [@dispatch]. Multiple dispatch allows `ExpFamilyPCA.jl` to integrate fast symbolic differentiation [@symbolics], optimization [@optim], and numerically stable computation [@stable_exp] without requiring costly API conversions. As a result, `ExpFamilyPCA.jl` delivers speed, stability, and flexibility, with built-in support for most common distributions (§ [Supported Distributions](#supported-distributions)) and flexible constructors for custom distributions (§ [Custom Distributions](#supported-distributions)).
+The absence of a general EPCA library likely stems from the limited interoperability between fast symbolic differentiation and optimization libraries in popular languages like Python and C. Julia, by contrast, uses multiple dispatch which promotes high levels of generic code reuse [@dispatch]. Multiple dispatch allows `ExpFamilyPCA.jl` to integrate fast symbolic differentiation [@symbolics], optimization [@optim], and numerically stable computation [@stable_exp] without requiring costly API conversions.[^1] As a result, `ExpFamilyPCA.jl` delivers speed, stability, and flexibility, with built-in support for most common distributions (§ [Supported Distributions](#supported-distributions)) and flexible constructors for custom distributions (§ [Custom Distributions](#supported-distributions)).
+
+[^1]: Symbolic differentiation is essential for flexibly specifying the EPCA objective (see [documentation](https://sisl.github.io/ExpFamilyPCA.jl/stable/math/objectives/#2.-Using-F-and-f)). While numeric differentiation is faster, symbolic differentiation is performed only once to generate a closed form for the optimizer (e.g., `Optim.jl` [@optim]), making it more efficient in practice. `LogExpFunctions.jl` (which implements ideas from @stable_exp) mitigates overflow and underflow in exponential and logarithmic operations.
 
 ## Principal Component Analysis
 

src/epca.jl

@@ -121,7 +121,7 @@ function fit!(
         steps_per_print,
         epca.options
     )
-    epca.V[:] = V  # TODO: delete this line?
+    epca.V[:] = V
     return A
 end