\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\bbl@cs{beforestart}
\bibstyle{plainnat}
\citation{yu2021icml_wspn}
\citation{whittle1953analysis}
\providecommand \oddpage@label [2]{}
\babel@aux{american}{}
\@writefile{toc}{\contentsline {section}{\numberline {A}Whittle likelihood and Whittle networks}{1}{appendix.A}\protected@file@percent }
\newlabel{app:wnet}{{A}{1}{Whittle likelihood and Whittle networks}{appendix.A}{}}
\newlabel{app:wnet@cref}{{[appendix][1][2147483647]A}{[1][1][]1}}
\newlabel{eq:complex_normal}{{2}{1}{Whittle likelihood and Whittle networks}{equation.A.2}{}}
\newlabel{eq:complex_normal@cref}{{[equation][2][2147483647]2}{[1][1][]1}}
\newlabel{eq:spectral_density_matrix}{{3}{1}{Whittle likelihood and Whittle networks}{equation.A.3}{}}
\newlabel{eq:spectral_density_matrix@cref}{{[equation][3][2147483647]3}{[1][1][]1}}
\citation{wolter2020sequence}
\citation{wolter2020sequence}
\citation{vaswani2017attention}
\citation{yang2020complex}
\newlabel{eq:whitle_likelihood_0}{{4}{2}{Whittle likelihood and Whittle networks}{equation.A.4}{}}
\newlabel{eq:whitle_likelihood_0@cref}{{[equation][4][2147483647]4}{[1][1][]2}}
\@writefile{toc}{\contentsline {section}{\numberline {B}Training Procedure}{2}{appendix.B}\protected@file@percent }
\newlabel{app:train}{{B}{2}{Training Procedure}{appendix.B}{}}
\newlabel{app:train@cref}{{[appendix][2][2147483647]B}{[1][2][]2}}
\@writefile{toc}{\contentsline {section}{\numberline {C}Short Time Fourier Transform}{2}{appendix.C}\protected@file@percent }
\newlabel{app:STFT}{{C}{2}{Short Time Fourier Transform}{appendix.C}{}}
\newlabel{app:STFT@cref}{{[appendix][3][2147483647]C}{[1][2][]2}}
\@writefile{toc}{\contentsline {section}{\numberline {D}Improving Spectral RNN}{2}{appendix.D}\protected@file@percent }
\newlabel{app:SRNN}{{D}{2}{Improving Spectral RNN}{appendix.D}{}}
\newlabel{app:SRNN@cref}{{[appendix][4][2147483647]D}{[1][2][]2}}
\@writefile{toc}{\contentsline {section}{\numberline {E}Conceiving the Spectral Transformer (STransformer)}{2}{appendix.E}\protected@file@percent }
\newlabel{app:STransformer}{{E}{2}{Conceiving the Spectral Transformer (STransformer)}{appendix.E}{}}
\newlabel{app:STransformer@cref}{{[appendix][5][2147483647]E}{[1][2][]2}}
\citation{yang2020complex}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Predictive Whittle networks training procedure together with the Whittle forecasting loss{} (here denoted as ``WFL'') allows to gauge the predictive likelihoods provided by the Whittle PC to guide the training towards more accurate neural spectral forecasting.\relax }}{3}{figure.caption.1}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:Training}{{1}{3}{Predictive Whittle networks training procedure together with the \lossname {} (here denoted as ``WFL'') allows to gauge the predictive likelihoods provided by the Whittle PC to guide the training towards more accurate neural spectral forecasting.\relax }{figure.caption.1}{}}
\newlabel{fig:Training@cref}{{[figure][1][2147483647]1}{[1][2][]3}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Forecasting accuracy (in MSE) of four different SRNN architecture proposals on the \textit  {Power} data set. Results are averaged over five runs with different seeds (best results in bold). Adding residual connections, dropout with $p = 0.1$, and a second layer is beneficial, thus, we will use this architecture for predictive Whittle networks{}.\relax }}{3}{table.caption.2}\protected@file@percent }
\newlabel{tbl:SRNNCMP}{{1}{3}{Forecasting accuracy (in MSE) of four different SRNN architecture proposals on the \textit {Power} data set. Results are averaged over five runs with different seeds (best results in bold). Adding residual connections, dropout with $p = 0.1$, and a second layer is beneficial, thus, we will use this architecture for \modelname {}.\relax }{table.caption.2}{}}
\newlabel{tbl:SRNNCMP@cref}{{[table][1][2147483647]1}{[1][2][]3}}
\@writefile{toc}{\contentsline {section}{\numberline {F}Whittle Einsum Networks (WEin) Implementation}{3}{appendix.F}\protected@file@percent }
\newlabel{app:wein}{{F}{3}{Whittle Einsum Networks (WEin) Implementation}{appendix.F}{}}
\newlabel{app:wein@cref}{{[appendix][6][2147483647]F}{[1][3][]3}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.1}Leaf Distributions}{3}{subsection.F.1}\protected@file@percent }
\citation{PeharzLVS00BKG20}
\citation{sato1999fast}
\citation{yu2021icml_wspn}
\citation{nielsen2009statistical}
\citation{peharz2016latent}
\citation{de2011strict}
\citation{pourahmadi2007simultaneous,li2019expectation}
\citation{higham1990analysis}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces A comparison of the SRNN operating in the real and in the complex space on \textit  {Power} and \textit  {Retail} data sets. When operating in the complex space, SRNN requires longer training times (in seconds) while providing only a marginal improvement in terms of accuracy in MSE (best results in bold).\relax }}{4}{table.caption.3}\protected@file@percent }
\newlabel{tbl:CSRNN}{{2}{4}{A comparison of the SRNN operating in the real and in the complex space on \textit {Power} and \textit {Retail} data sets. When operating in the complex space, SRNN requires longer training times (in seconds) while providing only a marginal improvement in terms of accuracy in MSE (best results in bold).\relax }{table.caption.3}{}}
\newlabel{tbl:CSRNN@cref}{{[table][2][2147483647]2}{[1][2][]4}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Preliminary experiments on the \textit  {Power} data set show that complex modeling is advantageous for transformer architectures (best results in bold). Compared to non-complex modeling, our complex STransformer improves forecasting accuracy while requiring a moderate amount of additional time for training. The ``emulated'' complex STransformer is less accurate than the complex STransformer and requires considerable additional time for training. \relax }}{4}{table.caption.4}\protected@file@percent }
\newlabel{tbl:STCMP}{{3}{4}{Preliminary experiments on the \textit {Power} data set show that complex modeling is advantageous for transformer architectures (best results in bold). Compared to non-complex modeling, our complex STransformer improves forecasting accuracy while requiring a moderate amount of additional time for training. The ``emulated'' complex STransformer is less accurate than the complex STransformer and requires considerable additional time for training. \relax }{table.caption.4}{}}
\newlabel{tbl:STCMP@cref}{{[table][3][2147483647]3}{[1][3][]4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {F.2}Leaf Layer Optimization}{4}{subsection.F.2}\protected@file@percent }
\newlabel{eq:LeafOptim}{{14}{4}{Leaf Layer Optimization}{equation.F.14}{}}
\newlabel{eq:LeafOptim@cref}{{[equation][14][2147483647]14}{[1][4][]4}}
\citation{sato1999fast}
\citation{PeharzLVS00BKG20}
\citation{PeharzLVS00BKG20}
\citation{wolter2020sequence}
\citation{makridakis2020m4}
\newlabel{eq:PSD}{{19}{5}{Leaf Layer Optimization}{equation.F.19}{}}
\newlabel{eq:PSD@cref}{{[equation][19][2147483647]19}{[1][5][]5}}
\@writefile{toc}{\contentsline {section}{\numberline {G}Data sets}{5}{appendix.G}\protected@file@percent }
\newlabel{app:datasets}{{G}{5}{Data sets}{appendix.G}{}}
\newlabel{app:datasets@cref}{{[appendix][7][2147483647]G}{[1][5][]5}}
\@writefile{toc}{\contentsline {section}{\numberline {H}Alternative visualization of LLRS}{5}{appendix.H}\protected@file@percent }
\newlabel{app:llrs}{{H}{5}{Alternative visualization of LLRS}{appendix.H}{}}
\newlabel{app:llrs@cref}{{[appendix][8][2147483647]H}{[1][5][]5}}
\@writefile{toc}{\contentsline {section}{\numberline {I}Correlation Error}{5}{appendix.I}\protected@file@percent }
\newlabel{app:ce}{{I}{5}{Correlation Error}{appendix.I}{}}
\newlabel{app:ce@cref}{{[appendix][9][2147483647]I}{[1][5][]5}}
\citation{papamakarios2017masked}
\citation{shao2020cspn}
\citation{chung2014empirical}
\citation{SALINAS20201181}
\citation{oreshkin2019n}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces An alternative visualization of the LLRS for long-range predictions on Power and Retail data sets. \relax }}{6}{figure.caption.5}\protected@file@percent }
\newlabel{fig:Alter_LLRS}{{2}{6}{An alternative visualization of the LLRS for long-range predictions on Power and Retail data sets. \relax }{figure.caption.5}{}}
\newlabel{fig:Alter_LLRS@cref}{{[figure][2][2147483647]2}{[1][5][]6}}
\@writefile{toc}{\contentsline {section}{\numberline {J}Experimental Setting and Model Capacity}{6}{appendix.J}\protected@file@percent }
\newlabel{app:capacity}{{J}{6}{Experimental Setting and Model Capacity}{appendix.J}{}}
\newlabel{app:capacity@cref}{{[appendix][10][2147483647]J}{[1][6][]6}}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces Test correlation error (lower is better) for different architectures modeling the time series in the time domain (denoted with ``\textit  {Time}'') or in the spectral domain. A lower score indicates a stronger correlation between CWLL and MSE. The results indicate that predictive Whittle networks{} can distinguish between ``good'' and ``bad'' predictions. Besides, modeling in the spectral domain generally outperforms modeling in the time domain w.r.t. the correlation error, in particular for MAF, where it considerably improves parameter efficiency. Furthermore, for smaller model sizes, predictive Whittle networks{} achieve the best scores, while MAF is better for models with larger capacity. \relax }}{7}{table.caption.6}\protected@file@percent }
\newlabel{tbl:CE2}{{4}{7}{Test correlation error (lower is better) for different architectures modeling the time series in the time domain (denoted with ``\textit {Time}'') or in the spectral domain. A lower score indicates a stronger correlation between CWLL and MSE. The results indicate that \modelname {} can distinguish between ``good'' and ``bad'' predictions. Besides, modeling in the spectral domain generally outperforms modeling in the time domain w.r.t. the correlation error, in particular for MAF, where it considerably improves parameter efficiency. Furthermore, for smaller model sizes, \modelname {} achieve the best scores, while MAF is better for models with larger capacity. \relax }{table.caption.6}{}}
\newlabel{tbl:CE2@cref}{{[table][4][2147483647]4}{[1][6][]7}}
\@writefile{lot}{\contentsline {table}{\numberline {5}{\ignorespaces Model capacity in \textbf  {thousands} of trainable parameters for each model for N-Beats and predictive Whittle networks{}. \relax }}{7}{table.caption.7}\protected@file@percent }
\newlabel{tbl:ModelCapacity}{{5}{7}{Model capacity in \textbf {thousands} of trainable parameters for each model for N-Beats and \modelname {}. \relax }{table.caption.7}{}}
\newlabel{tbl:ModelCapacity@cref}{{[table][5][2147483647]5}{[1][6][]7}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Whittle PCs can also be employed for forecasting via MPE queries. Predictions with the LLRS from CWSPN and WEin on \textit  {Power} are on the left and for \textit  {Retail} on the right. The context has been cut for clarity. The predictions computed with CWSPN are more accurate given its more discriminative nature. \relax }}{7}{figure.caption.8}\protected@file@percent }
\newlabel{fig:DataExampleMPE}{{3}{7}{Whittle PCs can also be employed for forecasting via MPE queries. Predictions with the LLRS from CWSPN and WEin on \textit {Power} are on the left and for \textit {Retail} on the right. The context has been cut for clarity. The predictions computed with CWSPN are more accurate given its more discriminative nature. \relax }{figure.caption.8}{}}
\newlabel{fig:DataExampleMPE@cref}{{[figure][3][2147483647]3}{[1][6][]7}}
\@writefile{toc}{\contentsline {section}{\numberline {K}Whittle PC Predictions via MPE}{7}{appendix.K}\protected@file@percent }
\newlabel{app:wpc}{{K}{7}{Whittle PC Predictions via MPE}{appendix.K}{}}
\newlabel{app:wpc@cref}{{[appendix][11][2147483647]K}{[1][7][]7}}
\bibdata{yu_296.bib}
\bibcite{chung2014empirical}{{1}{2014}{{Chung et~al.}}{{Chung, Gulcehre, Cho, and Bengio}}}
\bibcite{de2011strict}{{2}{2011}{{De~Iaco et~al.}}{{De~Iaco, Myers, and Posa}}}
\bibcite{higham1990analysis}{{3}{1990}{{Higham}}{{}}}
\bibcite{li2019expectation}{{4}{2019}{{Li and Au}}{{}}}
\bibcite{makridakis2020m4}{{5}{2020}{{Makridakis et~al.}}{{Makridakis, Spiliotis, and Assimakopoulos}}}
\bibcite{nielsen2009statistical}{{6}{2009}{{Nielsen and Garcia}}{{}}}
\bibcite{oreshkin2019n}{{7}{2019}{{Oreshkin et~al.}}{{Oreshkin, Carpov, Chapados, and Bengio}}}
\bibcite{papamakarios2017masked}{{8}{2017}{{Papamakarios et~al.}}{{Papamakarios, Pavlakou, and Murray}}}
\bibcite{peharz2016latent}{{9}{2016}{{Peharz et~al.}}{{Peharz, Gens, Pernkopf, and Domingos}}}
\bibcite{PeharzLVS00BKG20}{{10}{2020}{{Peharz et~al.}}{{Peharz, Lang, Vergari, Stelzner, Molina, Trapp, den Broeck, Kersting, and Ghahramani}}}
\bibcite{pourahmadi2007simultaneous}{{11}{2007}{{Pourahmadi et~al.}}{{Pourahmadi, Daniels, and Park}}}
\bibcite{SALINAS20201181}{{12}{2020}{{Salinas et~al.}}{{Salinas, Flunkert, Gasthaus, and Januschowski}}}
\bibcite{sato1999fast}{{13}{1999}{{Sato}}{{}}}
\bibcite{shao2020cspn}{{14}{2020}{{Shao et~al.}}{{Shao, Molina, Vergari, Stelzner, Peharz, Liebig, and Kersting}}}
\bibcite{vaswani2017attention}{{15}{2017}{{Vaswani et~al.}}{{Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser, and Polosukhin}}}
\bibcite{whittle1953analysis}{{16}{1953}{{Whittle}}{{}}}
\bibcite{wolter2020sequence}{{17}{2020}{{Wolter et~al.}}{{Wolter, Gall, and Yao}}}
\bibcite{yang2020complex}{{18}{2020}{{Yang et~al.}}{{Yang, Ma, Li, Tsai, and Salakhutdinov}}}
\bibcite{yu2021icml_wspn}{{19}{2021}{{Yu et~al.}}{{Yu, Ventola, and Kersting}}}
\newlabel{LastPage}{{}{8}{}{page.8}{}}
\xdef\lastpage@lastpage{8}
\xdef\lastpage@lastpageHy{8}
