and where $\Phi(x)$ denotes the normal cumulative distribution function. Note
that, from (33), $p(y\mid b,\sigma^{2})=N(y;b,\sigma^{2})$. Since
$\Phi(t+y)+\Phi(t-y)$ is non-increasing in $y\in(0,\infty)$ for any $t\geq 0$,
$p(y\mid\sigma^{2},t)$ is also non-increasing in $y\in(0,\infty)$ for any
$t\geq 0$. This implies that $p(y\mid\sigma^{2},t)$ is unimodal with a mode at
zero, and therefore $p(y\mid f,\sigma^{2})$ must also be unimodal with a mode
at zero since it is a mixture of unimodal distributions that all have modes at
zero.

From (66), which was obtained by applying Tweedie’s formula, we have that

$S_{f,\sigma}(y)=y+\sigma^{2}\ell_{\mathrm{NM}}^{\prime}(y;f,\sigma)\leq y,$

in which the inequality is obtained by noting that
$\ell_{\mathrm{NM}}(y;f,\sigma)$ is non-increasing in $y\in(0,\infty)$. And
since $\ell_{\mathrm{NM}}(y;f,\sigma^{2})$ is symmetric about zero, the
shrinkage operator must be an odd function; i.e.,
$S_{f,\sigma}(y)=-S_{f,\sigma}(-y)$.

It remains to show that $S_{f,\sigma}(y)$ is non-decreasing on
$y\in\mathbb{R}_{+}$. Since $p(y\mid b,\sigma^{2})=N(y;b,\sigma^{2})$ and
$p(b\mid t)$ is the uniform distribution on interval $[-t,t]$, the posterior
density is truncated normal:

$\displaystyle p(b\mid y,\sigma^{2},t)$ $\displaystyle\propto p(y\mid
b,\sigma^{2})\,p(b\mid t)$ $\displaystyle=N_{[-t,t]}(b;y,s^{2})$

where $N_{[-t,t]}(x;\mu,s^{2})=N(x;\mu,s^{2})\,\mathbb{I}\\{|x|<t\\}$ denotes
the probability density of the normal distribution with mean $\mu$ and
variance $s^{2}$ truncated to the interval $x\in[-t,t]$. The expected value of
the truncated normal,

$\mathbb{E}[X]=\mu+s\times\frac{N(-t;\mu,s^{2})-N(t;\mu,s^{2})}{\Phi((t-\mu)/s)-\Phi(-(t+\mu)/s)},$

is non-decreasing with respect to $\mu$, for all $\mu\in\mathbb{R}$, $t>0$. To
show this, the derivative of the expected value with respect to $\mu$ is
always positive:
$\frac{\partial}{\partial\mu}\mathbb{E}[X]=\mathrm{Var}(X)/s^{2}>0$.
Therefore, $S_{f,\sigma}$ is a mixture of non-decreasing functions on
$\mathbb{R}_{+}$.

## References

  * Amir et al. (2021) T. Amir, R. Basri, and B. Nadler.  The Trimmed Lasso: sparse recovery guarantees and practical optimization by the generalized soft-min penalty.  _SIAM Journal on Mathematics of Data Science_ , 3(3):900–929, 2021.
  * Anirban Bhattacharya and Dunson (2015) N. S. P. Anirban Bhattacharya, Debdeep Pati and D. B. Dunson.  Dirichlet–Laplace priors for optimal shrinkage.  _Journal of the American Statistical Association_ , 110(512):1479–1490, 2015.
  * Bai et al. (2021) R. Bai, V. Ročková, and E. I. George.  Spike-and-slab meets LASSO: A review of the spike-and-slab LASSO.  In M. G. Tadesse and M. Vannucci, editors, _Handbook of Bayesian Variable Selection_ , pages 81–108. Chapman and Hall/CRC, Boca Raton, FL, 2021\.
  * Beck and Tetruashvili (2013) A. Beck and L. Tetruashvili.  On the convergence of block coordinate descent type methods.  _SIAM Journal on Optimization_ , 23(4):2037–2060, 2013.
  * Berger (1985) J. O. Berger.  _Statistical Decision Theory and Bayesian Analysis_.  Springer, New York, NY, 1985.
  * Bertsekas (1999) D. P. Bertsekas.  _Nonlinear Programming_.  Athena Scientific, Belmont, MA, second edition, 1999.
  * Bertsimas et al. (2017) D. Bertsimas, M. S. Copenhaver, and R. Mazumder.  The Trimmed Lasso: sparsity and robustness.  _arXiv_ , 1708.04527, 2017.
  * Bhadra et al. (2019) A. Bhadra, J. Datta, N. G. Polson, and B. Willard.  Lasso meets horseshoe: a survey.  _Statistical Science_ , 34(3):405–427, 2019.
  * Bishop (2006) C. Bishop.  _Pattern Recognition and Machine Learning_.  Springer, New York, NY, 2006.
  * Blei et al. (2003) D. M. Blei, A. Y. Ng, and M. I. Jordan.  Latent Dirichlet allocation.  _Journal of Machine Learning Research_ , 3:993–1022, 2003\.
  * Blei et al. (2017) D. M. Blei, A. Kucukelbir, and J. D. McAuliffe.  Variational inference: a review for statisticians.  _Journal of the American Statistical Association_ , 112(518):859–877, 2017.
  * Breheny and Huang (2011) P. Breheny and J. Huang.  Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection.  _Annals of Applied Statistics_ , 5(1):232–253, 2011.
  * Carbonetto and Stephens (2012) P. Carbonetto and M. Stephens.  Scalable variational inference for Bayesian variable selection in regression, and its accuracy in genetic association studies.  _Bayesian Analysis_ , 7(1):73–108, 2012.
  * Carbonetto et al. (2017) P. Carbonetto, X. Zhou, and M. Stephens.  varbvs: fast variable selection for large-scale regression.  _arXiv_ , 1709.06597, 2017.
  * Carlin and Louis (2000) B. P. Carlin and T. A. Louis.  Empirical Bayes: past, present and future.  _Journal of the American Statistical Association_ , 95(452):1286–1289, 2000.
  * Carvalho et al. (2010) C. M. Carvalho, N. G. Polson, and J. G. Scott.  The horseshoe estimator for sparse signals.  _Biometrika_ , 97(2):465–480, 2010.
  * Casella (2001) G. Casella.  Empirical Bayes Gibbs sampling.  _Biostatistics_ , 2(4):485–500, 2001.
  * Castillo and Van Der Vaart (2012) I. Castillo and A. Van Der Vaart.  Needles and straw in a haystack: posterior concentration for possibly sparse sequences.  _Annals of Statistics_ , 40(4):2069–2101, 2012\.
  * Chipman et al. (2001) H. Chipman, E. I. George, and R. E. McCulloch.  The practical implementation of Bayesian model selection.  In _Model Selection_ , volume 38 of _IMS Lecture Notes_ , pages 65–116. 2001.
  * Dempster et al. (1977) A. P. Dempster, N. M. Laird, and D. B. Rubin.  Maximum likelihood from incomplete data via the EM algorithm.  _Journal of the Royal Statistical Society, Series B_ , 39(1):1–22, 1977.
  * Dharmadhikari and Joag-Dev (1988) S. Dharmadhikari and K. Joag-Dev.  _Unimodality, convexity, and applications_.  Academic Press, Boston, MA, 1988.
  * Drugowitsch (2013) J. Drugowitsch.  Variational Bayesian inference for linear and logistic regression.  _arXiv_ , 1310.5438, 2013.
  * Efron (2008) B. Efron.  Microarrays, empirical Bayes and the two-groups model.  _Statistical Science_ , 32(1):1–22, 2008.
  * Efron (2011) B. Efron.  Tweedie’s formula and selection bias.  _Journal of the American Statistical Association_ , 106(496):1602–1614, 2011.
  * Efron (2019) B. Efron.  Bayes, oracle Bayes and empirical Bayes.  _Statistical Science_ , 34(2):177–201, 2019.
  * Efron and Morris (1973) B. Efron and C. Morris.  Stein’s estimation rule and its competitors—an empirical Bayes approach.  _Journal of the American Statistical Association_ , 68(341):117–130, 1973.
  * Fan and Li (2011) J. Fan and R. Li.  Variable selection via nonconcave penalized likelihood and its oracle properties.  _Journal of the American Statistical Association_ , 96(456):1348–1360, 2011.
  * Figueiredo (2003) M. A. T. Figueiredo.  Adaptive sparseness for supervised learning.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 25(9):1150–1159, 2003.
  * Friedman et al. (2007) J. Friedman, T. Hastie, H. Höfling, and R. Tibshirani.  Pathwise coordinate optimization.  _Annals of Applied Statistics_ , 1(2):302–332, 2007.
  * Friedman et al. (2010) J. Friedman, T. Hastie, and R. Tibshirani.  Regularization paths for generalized linear models via coordinate descent.  _Journal of Statistical Software_ , 33(1):1–22, 2010.
  * Fu (1998) W. J. Fu.  Penalized regressions: The Bridge versus the Lasso.  _Journal of Computational and Graphical Statistics_ , 7(3):397–416, 1998.
  * Gamazon et al. (2015) E. R. Gamazon, H. E. Wheeler, K. P. Shah, S. V. Mozaffari, K. Aquino-Michaels, R. J. Carroll, A. E. Eyler, J. C. Denny, D. L. Nicolae, N. J. Cox, and H. K. Im.  A gene-based association method for mapping traits using reference transcriptome data.  _Nature Genetics_ , 47(9):1091–1098, 2015.
  * Gelman et al. (2013) A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin.  _Bayesian Data Analysis_.  CRC Press, Boca Raton, FL, third edition, 2013.
  * George and Foster (2000) E. I. George and D. P. Foster.  Calibration and empirical Bayes variable selection.  _Biometrika_ , 87(4):731–747, 2000.
  * George and McCulloch (1993) E. I. George and R. E. McCulloch.  Variable selection via Gibbs sampling.  _Journal of the American Statistical Association_ , 88(423):881–889, 1993.
  * George and McCulloch (1997) E. I. George and R. E. McCulloch.  Approaches for Bayesian variable selection.  _Statistica Sinica_ , 9(2):339–373, 1997.
  * Gerard and Stephens (2020) D. Gerard and M. Stephens.  Empirical Bayes shrinkage and false discovery rate estimation, allowing for unwanted variation.  _Biostatistics_ , 21(1):15–32, 2020.
  * Ghahramani and Hinton (2000) Z. Ghahramani and G. E. Hinton.  Variational learning for switching state-space models.  _Neural Computation_ , 12(4):831–864, 2000.
  * Girolami (2001) M. Girolami.  A variational method for learning sparse and overcomplete representations.  _Neural Computation_ , 13(11):2517–2532, 2001\.
  * Griffin and Brown (2010) J. E. Griffin and P. J. Brown.  Inference with normal-gamma prior distributions in regression problems.  _Bayesian Analysis_ , 5(1):171–188, 2010.
  * GTEx Consortium (2017) GTEx Consortium.  Genetic effects on gene expression across human tissues.  _Nature_ , 550(7675):204–213, 2017.
  * Guan and Stephens (2011) Y. Guan and M. Stephens.  Bayesian variable selection regression for genome-wide association studies and other large-scale problems.  _Annals of Applied Statistics_ , 5(3):1780–1815, 2011.
  * Habier et al. (2011) D. Habier, R. L. Fernando, K. Kizilkaya, and D. J. Garrick.  Extension of the Bayesian alphabet for genomic selection.  _BMC Bioinformatics_ , 12:186, 2011.
  * Hans (2009) C. Hans.  Bayesian lasso regression.  _Biometrika_ , 96(4):835–845, 2009.
  * Hartley and Rao (1967) H. O. Hartley and J. N. K. Rao.  Maximum-likelihood estimation for the mixed analysis of variance model.  _Biometrika_ , 54(1–2):93–108, 1967.
  * Hastie et al. (2009) T. Hastie, R. Tibshirani, and J. Friedman.  _The Elements of Statistical Learning_.  Springer, New York, NY, second edition, 2009.
  * Hazimeh and Mazumder (2020) H. Hazimeh and R. Mazumder.  Fast best subset selection: coordinate descent and local combinatorial optimization algorithms.  _Operations Research_ , 68(5):1517–1537, 2020\.
  * Hoerl and Kennard (1970) A. E. Hoerl and R. W. Kennard.  Ridge regression: biased estimation for nonorthogonal problems.  _Technometrics_ , 12(1):55–67, 1970.
  * Jaakkola and Jordan (2000) T. S. Jaakkola and M. I. Jordan.  Bayesian parameter estimation via variational methods.  _Statistics and Computing_ , 10:25–37, 2000.
  * Javanmard and Montanari (2018) A. Javanmard and A. Montanari.  Debiasing the lasso: optimal sample size for Gaussian designs.  _Annals of Statistics_ , 46(6A):2593–2622, 2018\.
  * Johnstone and Silverman (2004) I. M. Johnstone and B. W. Silverman.  Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences.  _Annals of Statistics_ , 32(4):1594–1649, 2004\.
  * Johnstone and Silverman (2005) I. M. Johnstone and B. W. Silverman.  Empirical Bayes selection of wavelet thresholds.  _Annals of Statistics_ , pages 1700–1752, 2005.
  * Joo (2017) L. Joo.  _Bayesian lasso: an extension for genome-wide association study_.  PhD thesis, New York University, 2017.
  * Jordan et al. (1999) M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul.  An introduction to variational methods for graphical models.  _Machine Learning_ , 37(2):183–233, 1999.
  * Kim et al. (2020) Y. Kim, P. Carbonetto, M. Stephens, and M. Anitescu.  A fast algorithm for maximum likelihood estimation of mixture proportions using sequential quadratic programming.  _Journal of Computational and Graphical Statistics, forthcoming_ , 29(2):261–273, 2020.
  * Koenker and Mizera (2014) R. Koenker and I. Mizera.  Convex optimization, shape constraints, compound decisions, and empirical Bayes rules.  _Journal of the American Statistical Association_ , 109(506):674–685, 2014.
  * Kullback and Leibler (1951) S. Kullback and R. A. Leibler.  On information and sufficiency.  _Annals of Mathematical Statistics_ , 22(1):79–86, 1951.
  * Li and Lin (2010) Q. Li and N. Lin.  The Bayesian elastic net.  _Bayesian Analysis_ , 5(1):151–170, 2010.
  * Liang et al. (2008) F. Liang, R. Paulo, G. Molina, M. a. Clyde, and J. O. Berger.  Mixtures of $g$ priors for Bayesian variable selection.  _Journal of the American Statistical Association_ , 103(481):410–423, 2008.
  * Logsdon et al. (2010) B. A. Logsdon, G. E. Hoffman, and J. G. Mezey.  A variational Bayes algorithm for fast and accurate multiple locus genome-wide association analysis.  _BMC Bioinformatics_ , 11:58, 2010.
  * Lu and Stephens (2016) M. Lu and M. Stephens.  Variance adaptive shrinkage (vash): flexible empirical Bayes estimation of variances.  _Bioinformatics_ , 32(22):3428–3434, 2016.
  * Luo and Tseng (1992) Z. Q. Luo and P. Tseng.  On the convergence of the coordinate descent method for convex differentiable minimization.  _Journal of Optimization Theory and Applications_ , 72:7–35, 1992.
  * Marlin et al. (2011) B. M. Marlin, M. E. Khan, and K. P. Murphy.  Piecewise bounds for estimating Bernoulli-logistic latent Gaussian models.  In _Proceedings of the 28th International Conference on Machine Learning_ , pages 633–640, 2011.
  * Mazumder et al. (2011) R. Mazumder, J. H. Friedman, and T. Hastie.  Sparsenet: coordinate descent with nonconvex penalties.  _Journal of the American Statistical Association_ , 106(495):1125–1138, 2011.
  * McCullagh and Nelder (1989) P. McCullagh and J. A. Nelder.  _Generalized Linear Models_ , volume 37 of _Monographs on Statistics and Applied Probability_.  Chapman and Hall, New York, NY, second edition, 1989.
  * Meuwissen et al. (2001) T. H. Meuwissen, B. J. Hayes, and M. E. Goddard.  Prediction of total genetic value using genome-wide dense marker maps.  _Genetics_ , 157(4):1819–1829, 2001.
  * Meuwissen et al. (2009) T. H. E. Meuwissen, T. R. Solberg, R. Shepherd, and J. A. Woolliams.  A fast algorithm for BayesB type of prediction of genome-wide estimates of genetic value.  _Genetics Selection Evolution_ , 41:2, 2009.
  * Miller (2002) A. J. Miller.  _Subset Selection in Regression_.  Chapman and Hall/CRC, Boca Raton, FL, 2002.
  * Mitchell and Beauchamp (1988) T. J. Mitchell and J. J. Beauchamp.  Bayesian variable selection in linear regression.  _Journal of the American Statistical Association_ , 83(404):1023–1032, 1988.
  * Morris (1983) C. N. Morris.  Parametric empirical Bayes inference: theory and applications.  _Journal of the American Statistical Association_ , 78(381):47–55, 1983.
  * Moser et al. (2015) G. Moser, S. H. Lee, B. J. Hayes, M. E. Goddard, N. R. Wray, and P. M. Visscher.  Simultaneous discovery, estimation and prediction analysis of complex traits using a Bayesian mixture model.  _PLoS Genetics_ , 11(4):e1004969, 2015.
  * Nebebe and Stroud (1986) F. Nebebe and T. Stroud.  Bayes and empirical Bayes shrinkage estimation of regression coefficients.  _Canadian Journal of Statistics_ , 14(4):267–280, 1986.
  * Parikh and Boyd (2014) N. Parikh and S. Boyd.  Proximal algorithms.  _Foundations and Trends in Optimization_ , 1(3):127–239, 2014.
  * Park and Casella (2008) T. Park and G. Casella.  The Bayesian Lasso.  _Journal of the American Statistical Association_ , 103(482):681–686, 2008.
  * Perez and de los Campos (2014) P. Perez and G. de los Campos.  Genome-wide regression and prediction with the BGLR statistical package.  _Genetics_ , 198(2):483–495, 2014.
  * R Core Team (2019) R Core Team.  _R: a language and environment for statistical computing_.  R Foundation for Statistical Computing, Vienna, Austria, 2019.  URL https://www.r-project.org.
  * Ray and Szabó (2022) K. Ray and B. Szabó.  Variational Bayes for high-dimensional linear regression with sparse priors.  _Journal of the American Statistical Association_ , 117(539):1270–1281, 2022.
  * Ren et al. (2011) Q. Ren, S. Banerjee, A. O. Finley, and J. S. Hodges.  Variational Bayesian methods for spatial data analysis.  _Computational Statistics and Data Analysis_ , 55(12):3197–3217, 2011.
  * Robbins (1964) H. Robbins.  The empirical Bayes approach to statistical decision problems.  _Annals of Mathematical Statistics_ , 35(1):1–20, 1964.
  * Ročková and George (2018) V. Ročková and E. I. George.  The Spike-and-Slab LASSO.  _Journal of the American Statistical Association_ , 113(521):431–444, 2018.
  * Saul and Jordan (1996) L. K. Saul and M. I. Jordan.  Exploiting tractable substructures in intractable networks.  In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors, _Advances in Neural Information Processing Systems 8_ , pages 486–492. MIT Press, 1996.
  * Smyth (2004) G. K. Smyth.  Linear models and empirical Bayes methods for assessing differential expression in microarray experiments.  _Statistical Applications in Genetics and Molecular Biology_ , 3(1), 2004.
  * Stephens (2016) M. Stephens.  False discovery rates: a new deal.  _Biostatistics_ , 18(2):275–294, 2016.
  * Stigler (1984) S. M. Stigler.  Studies in the history of probability and statistics XL: Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation.  _Biometrika_ , 71(3):615–620, 1984.
  * Su et al. (2017) W. Su, M. Bogdan, and E. Candes.  False discoveries occur early on the lasso path.  _Annals of Statistics_ , 45(5):2133–2150, 2017\.
  * Sun and Stephens (2018) L. Sun and M. Stephens.  Solving the empirical Bayes normal means problem with correlated noise.  _arXiv_ , 1812.07488, 2018.
  * Tibshirani (1996) R. Tibshirani.  Regression shrinkage and selection via the lasso.  _Journal of the Royal Statistical Society, Series B_ , pages 267–288, 1996.
  * Tikhonov (1963) A. N. Tikhonov.  Solution of incorrectly formulated problems and the regularization method.  _Soviet Mathematics—Doklady_ , 4:501–504, 1963.
  * Tseng (2001) P. Tseng.  Convergence of a block coordinate descent method for nondifferentiable minimization.  _Journal of Optimization Theory and Applications_ , 109(3):475–494, 2001.
  * Urbut et al. (2019) S. M. Urbut, G. Wang, P. Carbonetto, and M. Stephens.  Flexible statistical methods for estimating and testing effects in genomic studies with multiple conditions.  _Nature Genetics_ , 51(1):187–195, 2019.
  * van de Wiel et al. (2019) M. A. van de Wiel, D. E. Te Beest, and M. M. Münch.  Learning from a lot: empirical Bayes for high-dimensional model-based prediction.  _Scandinavian Journal of Statistics_ , 46(1):2–25, 2019.
  * Wainwright and Jordan (2008) M. J. Wainwright and M. I. Jordan.  Graphical models, exponential families, and variational inference.  _Foundations and Trends in Machine Learning_ , 1(1–2):1–305, 2008.
  * Wang and Titterington (2005) B. Wang and D. M. Titterington.  Inadequacy of interval estimates corresponding to variational bayesian approximations.  In _Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics_ , pages 373–380, 2005.
  * Wang and Blei (2013) C. Wang and D. M. Blei.  Variational inference in nonconjugate models.  _Journal of Machine Learning Research_ , 14:1005–1031, 2013\.
  * Wang et al. (2020) G. Wang, A. Sarkar, P. Carbonetto, and M. Stephens.  A simple new approach to variable selection in regression, with application to genetic fine mapping.  _Journal of the Royal Statistical Society, Series B_ , 82(5):1273–1300, 2020.
  * Wang and Stephens (2021) W. Wang and M. Stephens.  Empirical Bayes matrix factorization.  _Journal of Machine Learning Research_ , 22(120):1–40, 2021.
  * Wright (2015) S. J. Wright.  Coordinate descent algorithms.  _Mathematical Programming_ , 151:3–34, 2015.
  * Wu and Lange (2008) T. T. Wu and K. Lange.  Coordinate descent algorithms for lasso penalized regression.  _Annals of Applied Statistics_ , 2(1):224–244, 2008.
  * Xing et al. (2021) Z. Xing, P. Carbonetto, and M. Stephens.  Flexible signal denoising via flexible empirical Bayes shrinkage.  _Journal of Machine Learning Research_ , 22(93):1–28, 2021.
  * You et al. (2014) C. You, J. T. Ormerod, and S. Müller.  On variational Bayes estimation and variational information criteria for linear regression models.  _Australian and New Zealand Journal of Statistics_ , 56(1):73–87, 2014.
  * Yuan and Lin (2005) M. Yuan and Y. Lin.  Efficient empirical Bayes variable selection and estimation in linear models.  _Journal of the American Statistical Association_ , 100(472):1215–1225, 2005.
  * Yun et al. (2019) J. Yun, P. Zheng, E. Yang, A. Lozano, and A. Aravkin.  Trimming the $\ell_{1}$ regularizer: statistical analysis, optimization, and applications to deep learning.  In K. Chaudhuri and R. Salakhutdinov, editors, _Proceedings of the 36th International Conference on Machine Learning_ , pages 7242–7251, 2019\.
  * Zabad et al. (2023) S. Zabad, S. Gravel, and Y. Li.  Fast and accurate Bayesian polygenic risk modeling with variational inference.  _American Journal of Human Genetics_ , 110(5):741–761, 2023.
  * Zeng et al. (2018) J. Zeng, R. de Vlaming, Y. Wu, M. R. Robinson, L. R. Lloyd-Jones, L. Yengo, C. X. Yap, A. Xue, J. Sidorenko, A. F. McRae, J. E. Powell, G. W. Montgomery, A. Metspalu, T. Esko, G. Gibson, N. R. Wray, P. M. Visscher, and J. Yang.  Signatures of negative selection in the genetic architecture of human complex traits.  _Nature Genetics_ , 50(5):746–753, 2018.
  * Zhang (2010) C.-H. Zhang.  Nearly unbiased variable selection under minimax concave penalty.  _Annals of Statistics_ , 38(2):894–942, 2010\.
  * Zhou et al. (2013) X. Zhou, P. Carbonetto, and M. Stephens.  Polygenic modeling with Bayesian sparse linear mixed models.  _PLoS Genetics_ , 9(2), 2013.
  * Zhu et al. (2019) A. Zhu, J. G. Ibrahim, and M. I. Love.  Heavy-tailed prior distributions for sequence count data: removing the noise and preserving large differences.  _Bioinformatics_ , 35(12):2084–2092, 2019.
  * Zou and Hastie (2005) H. Zou and T. Hastie.  Regularization and variable selection via the elastic net.  _Journal of the Royal Statistical Society, Series B_ , 67(2):301–320, 2005.


# Diophantine Equation $n(x^{4}+y^{4})=z^{4}+w^{4}$

Seiji Tomita and Oliver Couto

###### Abstract

In this paper, we proved that there are infinitely many parametric solutions
of $n(x^{4}+y^{4})=z^{4}+w^{4}.$

1\. Introduction

In 2016 Izadi and Nabardi[2] showed $x^{4}+y^{4}=2(u^{4}+v^{4})$ has
infinitely many integer solutions.
They used a specifc congruent number elliptic curve namely
$y^{2}=x^{3}-36x^{2}$.
In 2019 Janfada and Nabardi[3] showed that a necessary condition for $n$ to
have an integral solution for the equation $x^{4}+y^{4}=n(u^{4}+v^{4})$ and
gave a parametric solution.
They gave the numeric solutions for $n=41,136,313,1028,1201,3281,\cdots$.

In 2020 Ajai Choudhry[1], Iliya Bluskov and Alexander James showed
$(x_{1}^{4}+x_{2}^{4})(y_{1}^{4}+y_{2}^{4})=z_{1}^{4}+z_{2}^{4}$ has
infinitely many parametric solutions. They gave the numeric solutions for
$n=17,257,626,641,706,1921,\cdots$.

Inspired by Choudhry’s article, we showed if $n=(a^{4}+b^{4})/2$ then
$n(x^{4}+y^{4})=z^{4}+w^{4}$ has infinitely many parametric solutions.
We consider n be an integer. Using four different methods we showed
$n(x^{4}+y^{4})=z^{4}+w^{4}$ has infinitely many integer solutions.
First method is similar as Choudhry’s one, second method is using another
identity.
Moreover, we showed the third method through an example
$17(x^{4}+y^{4})=z^{4}+w^{4}$.
In addition, we showed the fourth method through an example
$97(x^{4}+y^{4})=z^{4}+w^{4}$.
Finally, we gave the numeric solutions for
$n=2,8,17,41,82,97,113,136,137,146,178,193,\cdots$ with $n<1000$.

2\. First method

Theorem 1 $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$ has infinitely many
parametric solutions,
where $(a,b)=(\frac{m^{2}-n^{2}}{2},\frac{m^{2}+n^{2}}{2})$, $m,n$ are
arbitrary integer and have same parity.

Proof.

We use Brahmagupta’s identity

$(A^{2}+B^{2})(C^{2}+D^{2})=(AC+BD)^{2}+(AD-BC)^{2}$

.

We obtain

$(x^{4}+y^{4})(a^{2}+b^{2})=(x^{2}a+y^{2}b)^{2}+(x^{2}b-y^{2}a)^{2}$

Hence we have to find the rational solution of $x^{2}a+y^{2}b=u^{2}$ and
$x^{2}b-y^{2}a=v^{2}$.

Parametric solution of first equation is
$(x,y)=(k^{2}-2mk+4a-3m^{2},-k^{2}-2mk+4a-m^{2}$) where $a+b=m^{2}$.
Substitute it to second equation, we obtain

$v^{2}=(m^{2}-2a)k^{4}-4k^{3}m^{3}+(4am^{2}-2m^{4})k^{2}+(-32am^{3}+32a^{2}m+12m^{5})k+48a^{2}m^{2}-32a^{3}-34am^{4}+9m^{6}$

Let $a=\frac{m^{2}-n^{2}}{2}$ and $U=\frac{1}{k}$, then we obtain

$V^{2}=(5m^{4}n^{2}+4n^{6})U^{4}+(4m^{5}+8mn^{4})U^{3}-2m^{2}n^{2}U^{2}-4m^{3}U+n^{2}$
(1)

Since equation $(1)$ has a point $Q(U,V)=(0,n)$, then equation $(1)$ is
birationally equivalent to an elliptic curve $E$.

$\displaystyle E:Y^{2}-4m^{3}YX/n+(8nm^{5}+16mn^{5})Y$
$\displaystyle=X^{3}-2m^{2}(n^{4}+2m^{4})X^{2}/(n^{2})+(-20m^{4}n^{4}-16n^{8})X$
$\displaystyle+104m^{6}n^{6}+32m^{2}n^{10}+80m^{10}n^{2}$

Transformation is given,

$\displaystyle U$ $\displaystyle=\frac{2n^{2}X-4m^{2}n^{4}-8m^{6}}{Yn}$
$\displaystyle V$
$\displaystyle=\frac{n^{4}X^{3}-6n^{6}X^{2}m^{2}-12X^{2}m^{6}n^{2}+28n^{8}m^{4}X+32n^{4}m^{8}X+32m^{12}X-16m^{9}Yn+16n^{9}mY+16n^{12}X-104n^{10}m^{6}-32n^{14}m^{2}-80n^{6}m^{10}}{Y^{2}n^{3}}$
$\displaystyle X$ $\displaystyle=\frac{2nV+2n^{2}-4m^{3}U}{U^{2}}$
$\displaystyle Y$
$\displaystyle=\frac{4n^{3}V+4n^{4}-8n^{2}m^{3}U-4n^{4}m^{2}U^{2}-8m^{6}U^{2}}{U^{3}n}$

The point corresponding to point $Q$ is
$P(X,Y)=(\frac{2m^{2}(n^{4}+2m^{4})}{n^{2}},\frac{-16m(-m^{8}+n^{8})}{n^{3}})$.
According to the Nagell-Luts theorem, since the point $P$ is a point of
infinite order then we can obtain infinitely many rational points on $E$. Thus
we can obtain infinitely many rational solutions of $(1)$.
Therefore $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$ has infinitely many
parametric solutions.
The proof is completed.

When $m,n$ have opposite parity, problem links to
$h(x^{4}+y^{4})=2(z^{4}+w^{4})$ where $a^{2}+b^{2}=h/2$.

Example

$(a,b)=(\frac{m^{2}-n^{2}}{2},\frac{m^{2}+n^{2}}{2})$
$m,n$ are arbitrary integer and have same parity.

$2Q(U)=\frac{-2n^{2}m}{n^{4}+m^{4}}$

$\displaystyle x=-n^{4}+4m^{2}n^{2}+m^{4}$ $\displaystyle
y=n^{4}+4m^{2}n^{2}-m^{4}$ $\displaystyle z=(3n^{4}+m^{4})m$ $\displaystyle
w=(n^{4}+3m^{4})n$

$3Q(U)=\frac{2mn^{2}(n^{4}+3m^{4})}{n^{8}-8m^{4}n^{4}-m^{8}}$

$\displaystyle
x=m^{12}+12n^{2}m^{10}-19m^{8}n^{4}+40n^{6}m^{6}+19m^{4}n^{8}+12n^{10}m^{2}-n^{12}$
$\displaystyle
y=-m^{12}+12n^{2}m^{10}+19m^{8}n^{4}+40n^{6}m^{6}-19m^{4}n^{8}+12n^{10}m^{2}+n^{12}$
$\displaystyle z=m(m^{12}+41m^{8}n^{4}+27m^{4}n^{8}-5n^{12})$ $\displaystyle
w=n(-n^{12}-41m^{4}n^{8}-27m^{8}n^{4}+5m^{12})$

Numerical example: $(m,n)<10$

Table 1: Solutions of $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$ m | n | $a^{2}+b^{2}$ | x | y | z | w
---|---|---|---|---|---|---
3 | 1 | 41 | 29 | 11 | 63 | 61
4 | 2 | 136 | 31 | 1 | 76 | 98
5 | 1 | 313 | 181 | 131 | 785 | 469
5 | 3 | 353 | 361 | 89 | 1085 | 1467
6 | 4 | 776 | 209 | 79 | 774 | 1036
7 | 1 | 1201 | 649 | 551 | 4207 | 1801
7 | 3 | 1241 | 1021 | 139 | 4627 | 5463
7 | 5 | 1513 | 1669 | 781 | 7483 | 9785
8 | 2 | 2056 | 319 | 191 | 2072 | 1538
8 | 6 | 2696 | 751 | 401 | 3992 | 5094
9 | 1 | 3281 | 1721 | 1559 | 14769 | 4921
9 | 5 | 3593 | 3509 | 541 | 18981 | 25385
9 | 7 | 4481 | 5009 | 2929 | 30969 | 38647

3\. Second method

Theorem 2 $(x^{4}+y^{4})\frac{m^{4}+1}{2}=z^{4}+w^{4}$ has infinitely many
parametric solutions,
where $m$ is arbitrary odd number.

Proof.

We use an identity

$p(t+1)^{4}+p(t)^{4}=(t^{2}+at+b)^{2}+(ct^{2}+dt+e)^{2}$

where

$(a,b,c,e,p)=(d+2,\frac{1}{2d}+1,1+d,\frac{1}{2d},1+d+\frac{1}{2d^{2}})$

So, we look for the integer solutions $z^{2}=t^{2}+(d+2)t+\frac{1}{2d}+1$ and
$w^{2}=(1+d)t^{2}+dt+\frac{1}{2d}$. By parameterizing the first equation and
substituting the result to second equation, then we obtain quartic equation.

Let $d=m^{2}-1$ and $U=1/k$ then

$V^{2}=(4m^{6}+5m^{2})U^{4}+(-8m^{4}-4)U^{3}-2m^{2}U^{2}+4U+m^{2}$ (2)

This quartic equation is birationally equivalent to an elliptic curve below.

$E:Y^{2}-4YX/m+(8m+16m^{5})Y=X^{3}-2(m^{4}+2)X^{2}/(m^{2})+(-20m^{4}-16m^{8})X+104m^{6}+32m^{10}+80m^{2}$

Transformation is given,

$\displaystyle U=\frac{2m^{2}X-4m^{4}-8}{Ym}$ $\displaystyle
V=\frac{m^{4}X^{3}-6m^{6}X^{2}-12X^{2}m^{2}+28m^{8}X+32m^{4}X+32X+16Ym-16m^{9}Y+16m^{12}X-32m^{14}-104m^{10}-80m^{6}}{Y^{2}m^{3}}$
$\displaystyle X=\frac{2mV+2m^{2}+4U}{U^{2}}$ $\displaystyle
Y=\frac{4m^{3}V+4m^{4}+8m^{2}U-4m^{4}U^{2}-8U^{2}}{U^{3}m}$

It has a point
$P(X,Y)=(\frac{2(m^{4}+2)}{m^{2}},\frac{-16(-1+m^{8})}{m^{3}})$.
According to the Nagell-Luts theorem, since the point $P$ is a point of
infinite order then we can obtain infinitely many rational points on $E$.
Thus we can obtain infinitely many rational solutions of $(2)$ using group
law.
Therefore $(x^{4}+y^{4})\frac{m^{4}+1}{2}=z^{4}+w^{4}$ has infinitely many
parametric solutions.
The proof is completed.

When $m$ is even number, problem links to
$(m^{4}+1)(x^{4}+y^{4})=2(z^{4}+w^{4})$.

Example

$n=(m^{4}+1)/2$
$m$ is odd number.

$\displaystyle x=m^{4}+4m^{2}-1$ $\displaystyle y=m^{4}-4m^{2}-1$
$\displaystyle z=3m^{4}+1$ $\displaystyle w=(m^{4}+3)m$

$\displaystyle x=m^{12}+12m^{10}-19m^{8}+40m^{6}+19m^{4}+12m^{2}-1$
$\displaystyle y=m^{12}-12m^{10}-19m^{8}-40m^{6}+19m^{4}-12m^{2}-1$
$\displaystyle z=5m^{12}-27m^{8}-41m^{4}-1$ $\displaystyle
w=m(m^{12}+41m^{8}+27m^{4}-5)$

4\. Example for $41(x^{4}+y^{4})=z^{4}+w^{4}$

We show a numerical example of first method.

Let $(a,b,m)=(4,5,3)$ then $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$ is reduced
to

$V^{2}=U^{4}-108U^{3}-18U^{2}+996U+409$

Quartic equation is birationally equivalent to an elliptic curve $E$.

$E:Y^{2}+XY+Y=X^{3}-X^{2}-27X+26$

Transformation is given,

$\displaystyle U$ $\displaystyle=\frac{4Y+29X-10}{X-46}$ $\displaystyle V$
$\displaystyle=\frac{16X^{3}-2208X^{2}-3392X+13744-9840Y}{(X-46)^{2}}$
$\displaystyle X$ $\displaystyle=\frac{V+U^{2}-54U+5}{32}$ $\displaystyle Y$
$\displaystyle=\frac{UV+U^{3}-83U^{2}+99U-29V+175}{128}$

The rank of elliptic curve $E$ is $1$ and has generator $P(X,Y)=(6,-11)$ using
SAGE.
We can find infinitely many rational points on the curve E using the group
law.
Thus the multiples nP, n = 2, 3, …give infinitely many points as follows.

Table 2: Solutions of $41(x^{4}+y^{4})=z^{4}+w^{4}$ nP | x | y | z | w
---|---|---|---|---
$2P$ | 29 | 11 | 63 | 61
$3P$ | 17909 | 5149 | 37623 | 38699
$4P$ | 229422601 | 214213319 | 663306603 | 282177719
$5P$ | 81840455152441 | -86237007592439 | 252933880274523 | 61172008172039

5\. Example for $17(x^{4}+y^{4})=z^{4}+w^{4}$

The previous methods can’t give the rational solution of
$17(x^{4}+y^{4})=z^{4}+w^{4}$. Hence we show the third method of giving the
rational solution to the equation $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$. We
consider the simultaneous equation as follows.

$\left\\{\begin{array}[]{l}z^{2}-w^{2}-5x^{2}-3y^{2}=2t(4x^{2}+w^{2}-y^{2})\\\
(w^{2}+z^{2}+5x^{2}+3y^{2})t=-(x^{2}+w^{2}+4y^{2})\end{array}\right.$

Eliminating $z^{2}$, we get

$(1+10t+8t^{2})x^{2}+(4+6t-2t^{2})y^{2}+(2t^{2}+1+2t)w^{2}=0$ (3)

Parametric solution of $(3)$ is
$(x,y,w)=(2(k-2)(k-3),(k-1)(3k-7),-(25+5k^{2}-22k)).$

$\displaystyle z^{2}$ $\displaystyle=\frac{-w^{2}+17y^{2}}{4}$
$\displaystyle=\frac{4(2k^{2}-5k+1)(4k^{2}-15k+13)}{(-5+k^{2})^{2}}$

Hence cosider the quartic equation $(4)$

$V^{2}=8U^{4}-50U^{3}+105U^{2}-80U+13$ (4)

Quartic equation $(4)$ has a rational point $Q(U,V)=(2,1)$, then this quartic
equation is birationally equivalent to an elliptic curve below.

$E:Y^{2}=X^{3}-91X+330$ with rank is $1$ and generator=$(7,-6)$.

Transformation is given,

$\displaystyle U$ $\displaystyle=\frac{6X-36+2Y}{Y+2X-12}$ $\displaystyle V$
$\displaystyle=\frac{-18X^{2}-114+X^{3}+91X}{(Y+2X-12)^{2}}$ $\displaystyle X$
$\displaystyle=\frac{2V+6-U^{2}}{U^{2}-4U+4}$ $\displaystyle Y$
$\displaystyle=\frac{12V-108+180U-90U^{2}-4VU+14U^{3}}{U^{3}-6U^{2}+12U-8}$

The rank of elliptic curve $E$ is $1$ and has generator $P(X,Y)=(7,-6)$ using
SAGE.
We can find infinitely many rational points on the curve E using the group
law.
Thus the multiples nP, n = 2, 3, …give infinitely many points as follows.

Table 3: Solutions of $17(x^{4}+y^{4})=z^{4}+w^{4}$ nP | x | y | z | w
---|---|---|---|---
$2P$ | 3120 | 1921 | 2242 | 6529
$3P$ | 18418554 | 88538885 | 176117272 | 95896333
$4P$ | 87733253643360 | 108376421998081 | 198203611434238 | 206237591201281
$5P$ | 12509563104278834954874 | 6446124521923428875525 | 3117838409641509334568 | 25836199364300466735373

6\. Example for $97(x^{4}+y^{4})=z^{4}+w^{4}$

The previous methods can’t give the rational solution of
$97(x^{4}+y^{4})=z^{4}+w^{4}$. Hence we show the fourth method of giving the
rational solution to the equation $(x^{4}+y^{4})(a^{2}+b^{2})=z^{4}+w^{4}$.
According to Richmond’s theorem[4], existence of solution for diophantine
equation $ax^{4}+by^{4}+cz^{4}+dw^{4}=0$ are known if $abcd$ is square number.
He proved other solution is derived from a known solution. Repeting this
process, we can obtain infinitely many integer solutions. We use a known
solution $(x,y,z,w)=(112,71,10,37)$ obtained by brute force as follows.
Let $x=pt+112,y=qt+71,z=rt+10,w=t+37$.
We obtain

$\displaystyle(q^{4}+p^{4}-97r^{4}-97)t^{4}+(-3880r^{3}+284q^{3}-14356+448p^{3})t^{3}$
$\displaystyle+(-796758-58200r^{2}+30246q^{2}+75264p^{2})t^{2}+(-19653364+5619712p-388000r+1431644q)t=0$

In order to set the coefficient of $t$ and $t^{2}$ to zero, we obtain

$\displaystyle q=\frac{-135744p}{42103}+\frac{491693}{42103}$ $\displaystyle
r=\frac{-22422}{2965}+\frac{7672p}{2965}$

Hence we obtain

$t=\frac{-1238771307200}{68835707869p-174887242544}$

Take p=1 then $(q,r)=(\frac{355949}{42103},\frac{-2950}{593})$.

Finally, we obtain
$(x,y,z,w)=(174887242544,240033770927,68026751110,68835707869).$

Thus we can obtain other integer solution by using this new solution as a
known solution.
According to Richmond’s theorem, all solutions of Table 4 have infinitely many
integer solutions.

Smallest: $Min\ n(x^{4}+y^{4})$

Brute force search range: $n<1000,(x,y,z,w)<50000$

$n\equiv 1,2,8,9\mod 16$ is to be searched.

Table 4: Solutions of $n(x^{4}+y^{4})=(z^{4}+w^{4})$ n | x | y | z | w
---|---|---|---|---
2 | 7 | 20 | 21 | 19
8 | 19 | 21 | 40 | 14
17 | 5 | 6 | 13 | 8
41 | 1 | 1 | 3 | 1
82 | 219 | 320 | 1011 | 247
97 | 10 | 37 | 112 | 71
113 | 1 | 2 | 6 | 5
136 | 1 | 1 | 4 | 2
137 | 29 | 5 | 99 | 31
146 | 1 | 2 | 7 | 3
178 | 1 | 2 | 7 | 5
193 | 18 | 43 | 159 | 80
226 | 1 | 8 | 31 | 7
241 | 1 | 2 | 8 | 1
257 | 4 | 15 | 52 | 49
313 | 1 | 1 | 5 | 1
328 | 7 | 3 | 30 | 8
337 | 15 | 34 | 147 | 26
353 | 1 | 1 | 5 | 3
386 | 1 | 2 | 9 | 1
401 | 1 | 2 | 9 | 4
433 | 8 | 13 | 53 | 50
466 | 2 | 43 | 181 | 151
482 | 5 | 6 | 31 | 7
521 | 9 | 1 | 43 | 1
562 | 10 | 11 | 61 | 7
577 | 175 | 188 | 929 | 848
578 | 24 | 37 | 187 | 85
593 | 1 | 2 | 10 | 3
626 | 16 | 29 | 141 | 97
641 | 16 | 19 | 97 | 78
673 | 161 | 26 | 820 | 139
706 | 17 | 28 | 149 | 13
712 | 15 | 13 | 86 | 36
761 | 7 | 3 | 37 | 11
776 | 1 | 1 | 6 | 4
802 | 1 | 10 | 53 | 19
857 | 7 | 5 | 39 | 23
866 | 1 | 2 | 11 | 3
881 | 7 | 31 | 169 | 9
898 | 1 | 2 | 11 | 5
953 | 2041 | 2021 | 12975 | 7999
977 | 3 | 10 | 56 | 11

7\. Final remarks

There is an interesting relationship between $8(19^{4}+21^{4})=14^{4}+40^{4}$
and $2(7^{4}+20^{4})=19^{4}+21^{4}$
Multiply both sides of $2(7^{4}+20^{4})=19^{4}+21^{4}$ by $2^{3}$, then we
obtain $14^{4}+40^{4}=8(19^{4}+21^{4})$.
Thus an equation $n(x^{4}+y^{4})=z^{4}+w^{4}$ can be transform
$(nx)^{4}+(ny)^{4}=n^{3}(z^{4}+w^{4})$.
We could not find a solution of $n(x^{4}+y^{4})=z^{4}+w^{4}$ for
$n=34,40,50,65,73,89$ whre $n<100$. They are not the form $(m^{4}+n^{4})/2$. I
don’t know if they don’t have a solution to begin with, or if they have a
large solution.

## References

  * [1] Ajai Choudhry, Iliya Bluskov and Alexander James, A quartic diophantine equation inspired by Brahmagupta’s identity, 2020, arXiv.org
  * [2] F. Izadi and K. Nabardi, Diophantine equation $X^{4}+Y^{4}=2(U^{4}+V^{4})$, Math. Slovaca 66 (2016),
  * [3] Ali S. Janfada , Kamran Nabardi, ON DIOPHANTINE EQUATION $x^{4}+y^{4}=n(u^{4}+v^{4})$, December 2019, Mathematica Slovaca 69(6)
  * [4] H. W. Richmond, On the Diophantine equation $F=ax^{4}+by^{4}+cz^{4}+dw^{4}$, the product $abcd$ being square number, J. Lond. Math. Soc., 19 (1944)