chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3cb06591727b2d95c70719da54c4a6909247aec2 | subsection | 2 | 39 | Introduction | In light of this, we propose to invert the task of photometric redshift estimation.
That is to say, having stated the possibility to determine the redshift of a galaxy based on its photometry, we want to build a method that allows us to investigate the parameter space and to extract the features to be used to achieve t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 540,
"openalex_id": "",
"raw": "D'Isanto, A. & Polsterer, K. L. 2018, , 609, A111",
"source_ref_id": "bd8862137af506ffc423233383c5a1ba93ee168c",
"start": 341
},
{
"arxiv_id": "",
"doi": "",
"e... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.01985020749270916,
-0.001473317272029817,
-0.05465291813015938,
0.01855330727994442,
-0.02900480106472969,
-0.013312303461134434,
0.043819982558488846,
-0.0025499353650957346,
0.013007150031626225,
0.03289550170302391,
-0.00881129503250122,
-0.0003909773949999362,
-0.016340946778655052,
... |
68c8d106fdedc84a7a7ac5415b1b98bc58498374 | subsection | 3 | 39 | Introduction | The proposed approach is very general and could be also used to solve many other tasks in astronomy, including both regression and classification problems.Outline:
In Sec. the methodology and models used to perform the experiments are described together with the statistical estimators used to evaluate the performance.... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.032964568585157394,
-0.0028023698832839727,
-0.04050368815660477,
0.0096604498103261,
0.00980543252080679,
-0.021503277122974396,
0.012254883535206318,
0.000012042178241244983,
0.02757730334997177,
0.021991640329360962,
-0.017092738300561905,
0.0414804145693779,
0.018527308478951454,
0.... |
dd80d4ed2cb27279f624916c7b2608491424c0b8 | subsection | 4 | 39 | Methods | The main purpose of this work is to build an efficient method capable of generating, handling and selecting the best features for photometric redshift estimation, even though the proposed method is also able to deal with any other task of regression or even classification.
We calculate thousands of feature combinations... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 627,
"openalex_id": "",
"raw": "D'Isanto, A. & Polsterer, K. L. 2018, , 609, A111",
"source_ref_id": "bd8862137af506ffc423233383c5a1ba93ee168c",
"start": 530
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.024876657873392105,
-0.007646138779819012,
-0.04938703402876854,
0.00007207601447589695,
-0.02718118391931057,
-0.00063956284429878,
0.013392188586294651,
0.01895509846508503,
0.00741721224039793,
0.06678543239831924,
-0.04138987511396408,
-0.010156696662306786,
-0.0186956487596035,
0.0... |
6be6c12fa9df3d63ac7a0a4886d127ad4bad32de | subsection | 5 | 39 | Regression models | As mentioned above, our method makes use of kNN and RF models, which are described in detail in the following subsections, while the details regarding the deep convolutional mixture density network (DCMDN) used to compare the results with an automatic features extraction based model can be found in . | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 301,
"openalex_id": "",
"raw": "D'Isanto, A. & Polsterer, K. L. 2018, , 609, A111",
"source_ref_id": "bd8862137af506ffc423233383c5a1ba93ee168c",
"start": 0
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.0007606957806274295,
-0.05860457569360733,
-0.03763512521982193,
-0.007791050709784031,
-0.03265983983874321,
-0.04209151491522789,
0.0366889052093029,
-0.02528848499059677,
0.010438939556479454,
0.038733962923288345,
-0.040962155908346176,
0.02844763733446598,
-0.029958536848425865,
0.... |
09803c0b39beb587aecfb5f4004b2e59c4da7022 | subsection | 6 | 39 | kNN | The kNN is a machine learning model used both for regression and classification tasks .
This model explores the feature space by estimating the k nearest points (or neighbours) belonging to the training sample with respect to each test item.
In our case the distance involved is calculated through a Euclidean metric.
In... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 88,
"openalex_id": "",
"raw": "Fix, E. & Hodges, J. L. 1951, US Air Force School of Aviation Medicine, Technical Report 4, 477+",
"source_ref_id": "db2fd1c245c36697887a5e9752f3ebf50282522e",
"start": 0
},
{
... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.07167286425828934,
0.013423399068415165,
-0.0582418330013752,
-0.0001252004731213674,
-0.008234137669205666,
0.0039682588540017605,
0.009577239863574505,
-0.012339758686721325,
0.015277797356247902,
0.048229608684778214,
-0.052838895469903946,
0.03574485704302788,
-0.02161174826323986,
... |
02af717b2226fe4fcec2decd66bc2186f0a232d5 | subsection | 7 | 39 | Random forest | The RF is one of the most popular ensemble-based machine learning models, and could be used for regression and classification tasks .
It is an ensemble of decision trees, where each tree is meant to partition the feature space in order to find the best split that minimizes the variance.
Each decision tree is built by a... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2530946",
"end": 134,
"openalex_id": "https://openalex.org/W1594031697",
"raw": "Breiman, L., Friedman, J., Olshen, R., & Stone, C. 1984, Classification and Regression Trees (Monterey, CA: Wadsworth and Brooks)",
"source_ref_i... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.03074236959218979,
-0.014707515947520733,
-0.04403100907802582,
0.033595383167266846,
0.011823988519608974,
-0.035578761249780655,
0.04512949287891388,
0.05028627812862396,
-0.0024639666080474854,
0.03756214305758476,
-0.07420887798070908,
-0.0026584903243929148,
-0.006751115899533033,
... |
8be0cac86ff5fba74661e29b380f08703d684f50 | subsection | 8 | 39 | Features selection strategy | The huge number of features evaluated, as described in Sec. , imposes the need to establish an efficient feature selection process.
In fact, in order to estimate a subset of the best f=10 featuresThe reason for selecting 10 features is discussed in Sec. REF and Fig. REF, starting with r=4,520 features, would imply, if... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 651,
"openalex_id": "",
"raw": "Mao, K. 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 34, 629, cited By 68",
"source_ref_id": "b86cb38610cbc9a6cd042e800acb7b146f1fb608",
"start": 561
... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.014563033357262611,
0.002944646868854761,
-0.037502288818359375,
0.016538845375180244,
-0.015592897310853004,
0.003890595631673932,
0.023450374603271484,
-0.0012301148381084204,
-0.011008096858859062,
0.05172203481197357,
-0.005900736432522535,
0.007090801373124123,
-0.031017964705824852,... |
03e590bdedf468d97e3f571c95c0c9892afff5b5 | subsection | 9 | 39 | GPU parallelization for kNN | The feature selection is done by parallelizing the experiments on a GPU cluster.
The massive use of GPUs proved to be mandatory in order to deal with such an amount of data, features, k values, and runs on randomly sampled data-sets.
Following and , the kNN algorithm has been parallelized by using GPUs.
Typically, GPU-... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 305,
"openalex_id": "",
"raw": "Heinermann, J., Kramer, O., Polsterer, K., & Gieseke, F. 2013, in KI, Vol. 8077",
"source_ref_id": "bdfddc190199c8b33d8104da20479ad76bdd9219",
"start": 234
},
{
"arxiv_id":... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.034088119864463806,
0.016876213252544403,
-0.016860954463481903,
0.0048980689607560635,
-0.031768783926963806,
-0.005584714002907276,
0.024002064019441605,
-0.0023612964432686567,
-0.009956355206668377,
0.03933713957667351,
-0.026443468406796455,
0.029708847403526306,
-0.00974273215979337... |
f897d6270b2d7828091a30043cb00c9cdffe2cea | subsection | 10 | 39 | Statistical estimators and cross validation | The results have been evaluated using the following set of statistical scores for the quantity \Delta z = (\mbox{{$z_{\rm spec}$}}-\mbox{{$z_{\rm phot}$}})/(1+\mbox{{$z_{\rm spec}$}}) expressing the estimation errorWe note that \Delta z denotes the normalized error in redshift estimation and not the usually used plain ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 769,
"openalex_id": "",
"raw": "Hersbach, H. 2000, Weather and Forecasting, 15, 559",
"source_ref_id": "a9fe533d41e7205fbece471e0ee7186f80d0519f",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"e... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.06670751422643661,
0.01174094993621111,
-0.06859949231147766,
0.0025747695472091436,
-0.019087625667452812,
-0.032407768070697784,
0.04574316740036011,
-0.00524871563538909,
0.007369562983512878,
0.020094646140933037,
-0.05706452578306198,
0.027052246034145355,
0.00741915125399828,
0.07... |
4405fb820bda34e7c36f9e60c9e2830e5045c6d4 | subsection | 11 | 39 | Data | In the following subsections the details about the data-set used and the feature combinations performed for the experiments are outlined.The SDSS object IDs and coordinates of the extracted quasars for the three catalogues are available as supplementary information, as ASCII files.
The tables are only available in elec... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.01634414680302143,
-0.0009995719883590937,
-0.09125863015651703,
-0.002413088921457529,
-0.035648856312036514,
-0.04450003057718277,
0.05533508583903313,
-0.04483576491475105,
0.03732752799987793,
0.05054324120283127,
0.014795191586017609,
0.03537416458129883,
-0.026660339906811714,
0.0... |
9cc6650969cd5a14194def9931b9c1bae63a02e3 | subsection | 12 | 39 | Data-sets | The experiments are based on quasar data extracted from the SDSS DR7 () and SDSS DR9 ().
Three catalogues have been retrieved for the experiments.
Moreover, images for the DCMDN experiments have been downloaded making use of Hierarchical Progressive Survey . | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 88,
"openalex_id": "",
"raw": "Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, , 182, 543",
"source_ref_id": "ef38e74f00fbd2a7678f78002bb6d3f32e88144e",
"start": 0
},
{
"arxiv_id":... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.03066840022802353,
0.010764455422759056,
-0.0336589515209198,
-0.04873376712203026,
-0.023726051673293114,
-0.01447975728660822,
0.020064152777194977,
-0.020857563242316246,
-0.013930472545325756,
0.02647247537970543,
-0.0077357604168355465,
0.011580754071474075,
-0.0461399219930172,
0.... |
d4f12a00497a75dd183404f8f147a925e42d785b | subsection | 13 | 39 | Catalogue DR7a | Catalogue DR7a is the most conservative with respect to the presence of bad data or problematic objects.
It is based on DR7 only, with clean photometry and no missing data; the query used is reported in Appendix .
Furthermore, to be more conservative, we checked the spectroscopic redshifts in two different data release... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.03296595439314842,
-0.010645256377756596,
-0.014285246841609478,
-0.0006481587188318372,
-0.02783791720867157,
-0.06507723778486252,
0.012079885229468346,
-0.03879604488611221,
0.022954070940613747,
0.016650859266519547,
0.018589135259389877,
0.004387829452753067,
0.0010349554941058159,
... |
e7efb7b298d00da99a8f20dad2349fead18be09e | subsection | 14 | 39 | Catalogue DR7b | Catalogue DR7b has been obtained using the same query used for Catalogue DR7a, but removing the image processing flags.
This has been done in order to verify if the presence of objects previously discarded by the use of these flags could affect the feature selection process.
The catalogue has been cleaned by removing a... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.0013838037848472595,
-0.02674908936023712,
-0.03646909445524216,
0.01928742043673992,
-0.036713238805532455,
-0.041321467608213425,
0.008636614307761192,
-0.013931499794125557,
-0.02105746977031231,
0.0009842078434303403,
0.02108798734843731,
-0.008422987535595894,
0.00206569186411798,
... |
cb8ca5b11ee998cff2c5ce5a7af84e78aa7ac3c0 | subsection | 15 | 39 | Catalogue DR7+9 | Catalogue DR7+9 has been prepared mixing quasars from DR7 and DR9 in order to perform the feature selection with a different and more complete redshift distribution.
The difference in the redshift distribution of the two catalogues can be seen from the histogram in Fig. REF .
The catalogue has been cleaned with the sam... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.02869817242026329,
-0.014997503720223904,
-0.027431853115558624,
0.007880173623561859,
-0.030086548998951912,
-0.018354015424847603,
-0.02020009607076645,
-0.0019872074481099844,
-0.002231317339465022,
0.044611088931560516,
0.014471141621470451,
-0.002511662198230624,
-0.03025437332689762... |
b8aa7438b1f40fbd1d4fe66c00b36ee2205a52dd | subsection | 16 | 39 | Classic features | In classic redshift estimation experiments for quasars and galaxies, as can be found in the literature , for SDSS data colours are mainly used as features.
To be comparable, we decided to use a set of ten features as our benchmark feature set.
Colours of the adjacent filterbands for the point spread function (PSF) and ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 155,
"openalex_id": "",
"raw": "D'Abrusco, R., Staiano, A., Longo, G., et al. 2007, , 663, 752",
"source_ref_id": "e9d66dbfa8c604ac1d5251df07113258b589152d",
"start": 0
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.02910933457314968,
-0.021511616185307503,
-0.08531414717435837,
-0.03463217616081238,
-0.017712755128741264,
-0.035700127482414246,
0.037897057831287384,
-0.02541727013885975,
0.043511439114809036,
0.05162787809967995,
-0.018017884343862534,
0.034601662307977676,
-0.0030951553490012884,
... |
cad43b7861b7d4e7e449203cfc3618ca9fe674ca | subsection | 17 | 39 | Combined features | For each of the three catalogues, the features concerning magnitudes and their errors, radii, ellipticities, and extinction are retrieved.
An overview of the features is shown in Table REF .
Magnitudes that have been corrected for extinction are denoted with an underline indicating that, for example, u_{model} is equiv... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 456,
"openalex_id": "",
"raw": "Gieseke, F., Polsterer, K. L., Oancea, C. E., & Igel, C. 2014, in 22th European Symposium on Artificial Neural Networks, ESANN 2014, Bruges, Belgium, April 23-25, 2014",
"source_ref_id": "a3d3... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.01290735974907875,
0.0022103472147136927,
-0.03747101128101349,
-0.0023591022472828627,
-0.027477724477648735,
-0.04409251734614372,
0.03240571171045303,
-0.0019757719710469246,
0.017911631613969803,
0.03292444720864296,
-0.02441108226776123,
0.027111558243632317,
-0.014844989404082298,
... |
e419e82e7e082b36ebf0554004e895ab2f3c165a | subsection | 18 | 39 | Experiments and results | The feature selection was performed applying the forward selection strategy, as described in Sec. REF , on the three catalogues.
The verification of the resulting feature sets was performed using the RF.
This algorithm is widely used in literature, and therefore the results obtained here can be easily compared to those... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.019933057948946953,
0.013667727820575237,
-0.0466122180223465,
-0.006414140574634075,
-0.026389170438051224,
-0.019414126873016357,
0.013286161236464977,
-0.0018048116471618414,
0.016575267538428307,
0.04258287325501442,
-0.019520964473485947,
0.02834279276430607,
-0.020833555608987808,
... |
7e83324049965baf4f508a1326422b448fce5274 | subsection | 19 | 39 | Experiment DR7a | The feature selection on the Catalogue DR7a produced 22 subsets of ten features each.
Only 20 features, of the initial 4,520, compose the tree.
The three features,\text{{$g_{psf}$}}/\text{{$u_{model}$}}
\text{{$i_{psf}$}}/\text{{$z_{model}$}}
z_{model}/z_{psf,}appear in all the possible branches.
For all presented fe... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1101/gr.092759.109",
"end": 1914,
"openalex_id": "https://openalex.org/W2158804744",
"raw": "Krzywinski, M. I., Schein, J. E., Birol, I., et al. 2009, Genome Research [e-print]",
"source_ref_id": "49533d25917a889dc2edd1a55a7b2b6078... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04257819429039955,
-0.0023540095426142216,
-0.04300549998879433,
-0.04712597280740738,
-0.009553386829793453,
-0.002657321747392416,
0.026767795905470848,
-0.018267419189214706,
0.03146818280220032,
0.0517653152346611,
-0.02405133843421936,
0.017840109765529633,
-0.0028652530163526535,
... |
126e0eca0cc973c52b136d7b63e1bf4616cdeb33 | subsection | 20 | 39 | Experiment DR7b | In the experiment performed with Catalogue DR7b, the proposed model selected 26 features generating 41 subset combinations.
Only the following two features appear in all the subsets:i_{psf}/i_{petro}
\text{{$g_{psf}$}}/\text{{$u_{model}$}}.From the RF validation runs, the subset reported in the third column of Table R... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.02863643877208233,
-0.01190769299864769,
-0.04296228662133217,
-0.03798867017030716,
-0.029582342132925987,
-0.040795862674713135,
0.03643250837922096,
-0.00021585554350167513,
0.02398320846259594,
0.03948380425572395,
-0.00527874706313014,
0.007727413903921843,
-0.014180910773575306,
0... |
e145e77ee5f2f34cdb3ad23bc2b19613d7c7022b | subsection | 21 | 39 | Experiment DR7b | The detailed feature selection results for this experiment and the chord diagram are also shown in Appendix .SELECTs.specObjID, p.objid, p.ra, p.dec, s.targetObjID, s.z, s.zErr,p.psfMag_u, p.psfMag_g, p.psfMag_r, p.psfMag_i, p.psfMag_z,p.psfMagErr_u, p.psfMagErr_g, p.psfMagErr_r, p.psfMagErr_i, p.psfMagErr_z,p.modelMag... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04515649750828743,
0.012845193035900593,
-0.055225178599357605,
-0.0036136640701442957,
-0.016796385869383812,
-0.03746768832206726,
0.008993160910904408,
-0.029961947351694107,
0.009580500423908234,
0.054767508059740067,
0.01574375294148922,
-0.004050354938954115,
-0.013890200294554234,
... |
8484e427717338a4c4a5d89c192b3873bdf2e806 | subsection | 22 | 39 | Experiment DR7+9 | The feature selected from the Catalogue DR7+9 are shown in Table REF . In Fig. REF a chord diagram is given to visualize the structure of the individual subsets.
In this experiment the model selected 14 individual features grouped in nine subsets.
Due to the different redshift distribution, different features are selec... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2352,
"openalex_id": "",
"raw": "Gneiting, T., Raftery, A. E., Westveld, A. H., & Goldman, T. 2005, Monthly Weather Review, 133, 1098",
"source_ref_id": "4fdd6407181c70987a64f472f33f1780df926461",
"start": 2230
}
... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04462729021906853,
0.004906865302473307,
-0.0550973042845726,
0.014758751727640629,
-0.019871659576892853,
0.001949773053638637,
0.02541191503405571,
-0.0038499434012919664,
0.020146382972598076,
0.05726456642150879,
-0.04447466880083084,
0.009378751739859581,
-0.01898643933236599,
0.04... |
c8434ea6561591ef99842a9064c81fd790980144 | subsection | 23 | 39 | Experiment DR7+9 | This is a confirmation of the quality and strength of the proposed method.SELECTm.objid, m.ra AS ra1, m.dec AS dec1,n.objid, n.distance,p.ra AS ra2, p.dec AS dec2,p.objid, p.ra, p.dec, p.psfMag_u, p.psfMag_g, p.psfMag_r, p.psfMag_i,p.psfMag_z,p.psfMagErr_u, p.psfMagErr_g, p.psfMagErr_r, p.psfMagErr_i,p.psfMagErr_z,p.mo... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.019556820392608643,
-0.011967492289841175,
-0.055497437715530396,
0.00028412305982783437,
-0.03151668608188629,
-0.003310319734737277,
0.013775201514363289,
-0.014942203648388386,
0.030799703672528267,
0.04710722342133522,
-0.01990768313407898,
0.007139307446777821,
-0.027641933411359787,... |
b5b82f99e24981e52762ae0051d8bf3e9232067d | subsection | 24 | 39 | Discussion | In the following subsections we discuss in detail the features found with the proposed method, the improvement in performance of the photometric redshift estimation models in comparison to the classic features, and the physical interpretation of the selected features. | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.030848825350403786,
0.003320215502753854,
-0.011167824268341064,
0.012464634142816067,
-0.021099863573908806,
-0.05257420986890793,
0.02251872792840004,
0.02721775695681572,
0.047997232526540756,
0.022305134683847427,
-0.034205276519060135,
-0.009977810084819794,
0.0023075593635439873,
... |
fed250be3fe9e51876234dbb7c405efc325b5dcd | subsection | 25 | 39 | Features | The results obtained from the feature selection process for the three experiments demonstrate that most of the information can be embedded in a limited number of features with respect to the initially generated amount of pairwise combinations.
The following four features have been selected and are in common between all... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.022475590929389,
0.009536486119031906,
-0.04299810901284218,
0.001234021270647645,
-0.00038861180655658245,
-0.02520683966577053,
0.05718839913606644,
-0.004196053836494684,
0.012611049227416515,
0.047300972044467926,
-0.019118746742606163,
-0.01968330703675747,
-0.0024470624048262835,
... |
aa2f093f90dd7bab383649e289e634238f1f004b | subsection | 26 | 39 | Features | This can be understood considering that the latter are ratios between magnitudes of the same filter where the contribution of the extinction correction tends to cancel out.Another relevant aspect in experiment DR7+9 is that all the 15 features in the tree are exclusively a composition of magnitudes and their errors.
Ne... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1347,
"openalex_id": "",
"raw": "Lupton, R. H., Gunn, J. E., & Szalay, A. S. 1999, , 118, 1406",
"source_ref_id": "da67bea31c05e68889a33cbae9cc4bfcb9d5b6a3",
"start": 1245
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.013614310882985592,
0.007700022310018539,
-0.06117282435297966,
-0.015590828843414783,
-0.02289402112364769,
-0.04542173817753792,
0.01614791713654995,
-0.01514821033924818,
0.02223772555589676,
0.046795379370450974,
-0.013095379807054996,
-0.017735235393047333,
-0.020955661311745644,
0... |
11d4905293746ac9a46ba497187141d2f004d7ec | subsection | 27 | 39 | Comparison of performance | Using the RF, the validation experiments were carried out on every feature set.
The second subset, indicated as Best_{10}, gave a slightly better performance than the others.
Even though we would not consider this as a substantial effect, we decided to choose this as our reference set.
It can be noticed from Fig. REF t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1909,
"openalex_id": "",
"raw": "D'Isanto, A. & Polsterer, K. L. 2018, , 609, A111",
"source_ref_id": "bd8862137af506ffc423233383c5a1ba93ee168c",
"start": 1744
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04521986097097397,
-0.009436060674488544,
-0.05724187195301056,
-0.00008849882578942925,
-0.036310143768787384,
-0.018978916108608246,
0.03945295512676239,
-0.003447938011959195,
0.03090939112007618,
0.051719069480895996,
-0.05693674460053444,
-0.0012910696677863598,
-0.01582084782421589,... |
e330c8d105cd3fa277264c3051b623efbb15a8a8 | subsection | 28 | 39 | Comparison of performance | The method captures the inherent structure of the physical properties of the sources, which is essential to provide good photometrically estimated redshifts for quasars.
[Table: Cross experiments performed with the RF, using the Best_{10} sets obtained from every experiment with all the three catalogues. The results ar... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.03871631622314453,
-0.008374805562198162,
-0.07529698312282562,
0.0014301239280030131,
0.004965390544384718,
-0.01618519052863121,
0.032767001539468765,
-0.023827772587537766,
-0.014095301739871502,
0.055648982524871826,
-0.05522185191512108,
0.01983105204999447,
-0.010609612800180912,
... |
220b527771a2bbedbb1b0f553c1bc60e7b630d96 | subsection | 29 | 39 | Physical interpretation | In contrast to deep learning models, feature-based approaches have the advantage of allowing an interpretation in a physical context.
Therefore the features selected by our approach are discussed in the following.
By analysing the importance of each feature of the Best_{10} set in smaller redshift bins, the contributio... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2530946",
"end": 472,
"openalex_id": "https://openalex.org/W1594031697",
"raw": "Breiman, L., Friedman, J., Olshen, R., & Stone, C. 1984, Classification and Regression Trees (Monterey, CA: Wadsworth and Brooks)",
"source_ref_i... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.017719706520438194,
-0.01528534572571516,
-0.08455398678779602,
0.0227105300873518,
-0.015002991072833538,
-0.02702980302274227,
0.013988037593662739,
-0.02008538693189621,
0.02423677407205105,
0.04468845948576927,
-0.05995091423392296,
0.010981334373354912,
-0.026800867170095444,
0.073... |
83cbfd3355b14e02682c3b9c026ea9498cbd50e0 | subsection | 30 | 39 | Physical interpretation | Using the well known relationz = \frac{\lambda _{observed}}{\lambda _{emitted}} - 1 = \frac{\lambda _{filter\ intersection}}{\lambda _{qso\ emission\ line}} - 1,it is possible to calculate the redshift at which a specific emission line becomes traceable when using a certain filter combination.
The proposed features cap... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2426,
"openalex_id": "",
"raw": "Wright, E. L. 2006, , 118, 1711",
"source_ref_id": "083629ec1f96e87de6d36ea69b9d61baa39c2c44",
"start": 2102
}
]
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04097864404320717,
-0.024700084701180458,
-0.06032373756170273,
-0.028331102803349495,
-0.014295230619609356,
-0.012113568373024464,
0.0027175431605428457,
-0.013837539590895176,
0.04189402610063553,
0.035852499306201935,
-0.02796494960784912,
0.0237999577075243,
0.009390303865075111,
0... |
42c04b044793dd39efb79df57d8831648416639a | subsection | 31 | 39 | Physical interpretation | As the selected features combine the strength of identifying line transitions as well as morphological characteristics, the resulting boost in performance of the photometric redshift estimation model can be well explained.
To explain the meaning of the selected features that use a combination of features extracted from... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.03174765035510063,
0.0030240400228649378,
-0.04084457457065582,
0.024848641827702522,
-0.018193844705820084,
-0.07185958325862885,
0.0023677183780819178,
0.03519715368747711,
0.034769780933856964,
0.0015644875820726156,
-0.05574154853820801,
-0.021154925227165222,
-0.02255914732813835,
... |
9d5eb138fd5e4956d26d21f6dfd28a2f0627d4ae | subsection | 32 | 39 | Conclusions | In this work a method to select the best features for photometric redshift estimation is proposed.
The features are calculated via a greedy forward selection approach, in which the features are selected from a set of 4,520 combinations based on the photometric and shape information stored in the SDSS DR7 and DR9 catalo... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.011662554927170277,
0.02117413654923439,
-0.07224834710359573,
0.0017934325151145458,
-0.030922170728445053,
-0.0415244922041893,
0.025170981884002686,
0.010617578402161598,
0.031852733343839645,
0.059800151735544205,
-0.02320306934416294,
0.008199638687074184,
-0.029549207538366318,
0.... |
81513795238809b70a06fe69eba44114bd250f23 | subsection | 33 | 39 | Conclusions | Both approaches are meant to establish an affordable and well-performing method to precisely predict photometric redshifts, in light of the upcoming missions and instruments in the near future.The authors gratefully acknowledge the support of the Klaus Tschira Foundation.
SC acknowledges support from the project “Quasa... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 552,
"openalex_id": "https://openalex.org/W3127445491",
"raw": "Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britt... | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.07373984158039093,
-0.007401453331112862,
-0.06592634320259094,
-0.015008926391601562,
-0.006703274790197611,
-0.01392541453242302,
0.007546429987996817,
-0.03543541207909584,
0.030170459300279617,
0.04782712087035179,
-0.03105558268725872,
0.007927948608994484,
0.0050780074670910835,
0... |
23d8ba517a10f80ddb2e79d0085e7e605f3f2f67 | subsection | 34 | 39 | Additional tables and figures | In this section, the additional tables for the features selection and the tree structure, together with the related chord diagrams for the experiments DR7a and DR7b are given.
A brief explanation of how to read a chord diagram follows. | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.0012168033281341195,
0.02284080907702446,
-0.02702142484486103,
-0.0155781339854002,
-0.024061426520347595,
-0.036160800606012344,
0.046902235597372055,
-0.0004987367428839207,
0.0031888638623058796,
0.026563692837953568,
-0.0026853589806705713,
-0.028669258579611778,
-0.01948411017656326... |
a99d6035de2968426c8ee55f88b719b166863f55 | subsection | 35 | 39 | Chord diagram: how to read | The chord diagram is a tool to visualize complex structures and relations in multidimensional data, which is arranged in a matrix shape.
The data are disposed in a circle and each element, in our case the features, is associated with a different colour.
The relations between the elements are expressed by ribbons which ... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.003433013102039695,
-0.008414696902036667,
-0.056331928819417953,
0.010558422654867172,
-0.01496030855923891,
-0.020430242642760277,
0.028150707483291626,
0.00049874052638188,
-0.010939868167042732,
0.029539169743657112,
-0.028410090133547783,
0.005275396630167961,
-0.003921263851225376,
... |
f8f64a2cdd067a27beb5861a73bdf41f8429b462 | subsection | 36 | 39 | Code | The code of the DCMDN model is available on the ASCL.http://www.ascl.net/ascl:1709.006.
[Table: Detailed feature branches obtained from the feature selection for the experiment DR7a. The 20th branch, indicated with the * symbol, is the best performing subset with respect to the experiments using the RF. The ratios and ... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.04200439900159836,
-0.007719376590102911,
-0.03626542538404465,
-0.06593713909387589,
-0.041790712624788284,
-0.02892381325364113,
0.018926400691270828,
-0.00031599646899849176,
0.032296985387802124,
0.04014228284358978,
-0.004205018747597933,
0.030022764578461647,
0.007276743184775114,
... |
5e04b709d421b710996e3fe40e7393188433ce30 | subsection | 37 | 39 | SDSS QSO query | In the following, the statements used to query the SDSS database are provided. | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
0.006360157858580351,
0.03769547492265701,
-0.09681785106658936,
0.0034013681579381227,
-0.0539335273206234,
-0.04468516260385513,
0.029820630326867104,
-0.0034414289984852076,
0.010629513300955296,
-0.0019524958916008472,
0.030415823683142662,
0.00557801453396678,
-0.04840892553329468,
0.... |
6589579449edbb22feb7d415d0b808704289d16f | subsection | 38 | 39 | Experiment DR7 | SELECTs.specObjID, p.objid, p.ra, p.dec, s.targetObjID, s.z, s.zErr,p.psfMag_u, p.psfMag_g, p.psfMag_r, p.psfMag_i, p.psfMag_z,p.psfMagErr_u, p.psfMagErr_g, p.psfMagErr_r, p.psfMagErr_i, p.psfMagErr_z,p.modelMag_u, p.modelMag_g, p.modelMag_r, p.modelMag_i, p.modelMag_z,p.modelMagErr_u, p.modelMagErr_g, p.modelMagErr_r,... | {
"cite_spans": []
} | 10.1051/0004-6361/201833103 | 1803.10032 | Return of the features. Efficient feature selection and interpretation
for photometric redshifts | [
"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
"Kai Lars Polsterer"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
-0.06112029030919075,
-0.02926328405737877,
-0.04812116548418999,
0.01687444932758808,
-0.02351132407784462,
-0.01820182427763939,
0.0005087318131700158,
-0.03600696101784706,
-0.010588487610220909,
0.019010454416275024,
0.0028054153081029654,
-0.012777893804013729,
-0.02442675642669201,
0... |
8f3e30eb9c6241b9e07f306ae9193dd40eb7026d | abstract | 0 | 18 | Abstract | Bayesian Additive Regression Trees (BART) is a fully Bayesian approach to
modeling with ensembles of trees. BART can uncover complex regression functions
with high dimensional regressors in a fairly automatic way and provide Bayesian
quantification of the uncertainty through the posterior. However, BART assumes
IID nor... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.018795501440763474,
-0.025965861976146698,
-0.03621795400977135,
0.04613441228866577,
-0.014584820717573166,
-0.028269531205296516,
0.0779280960559845,
0.0027136767748743296,
0.02489793673157692,
0.04256448522210121,
-0.07231386005878448,
0.01054195687174797,
0.007605160120874643,
0.041... | |
eec891f00f6eb089676c20cee99db9e3c630e641 | subsection | 1 | 18 | Introduction | Data analysts have long sought to uncover the information
in data without making strong assumptions about the nature of the underlying process.
Traditionally, asymptotic approaches to frequentist inference have been used.
The asymptotics often play the role of minimizing the assumptions needed to make an inference.
As ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 710,
"openalex_id": "",
"raw": "Dale Poirier. Intermediate Statistics and Econometrics, A Comparitive Approach. MIT, 1995.",
"source_ref_id": "583bea94012dc11830b0cecd8e8f57f03e465f91",
"start": 636
},
{
... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.01055703405290842,
-0.018688391894102097,
-0.0437232106924057,
0.04131278768181801,
-0.0037777249235659838,
-0.04207557812333107,
0.07475356757640839,
-0.0005377680063247681,
0.026392586529254913,
0.028192773461341858,
-0.050893448293209076,
0.029947195202112198,
0.007345682010054588,
0... | |
bff1deccd181d9dd1e1205f3eaf7760a60126489 | subsection | 2 | 18 | BART: Bayesian Additive Regression Trees | In this section we review the basic BART model.
is a fully Bayesian development of an ensemble of trees model in which the overall
conditional mean of a response given predictors is expressed as the sum of many trees.
The BART algorithm includes an effective Markov Chain Monte Carlo algorithm
which explores the complex... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1214/aos/1013203451",
"end": 517,
"openalex_id": "https://openalex.org/W1678356000",
"raw": "Jerome H Friedman. Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 29:1189–1232, 2001.",
"source_ref... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.02439573034644127,
-0.02943049557507038,
-0.03441949188709259,
0.0019700429402291775,
0.04247511550784111,
-0.029125358909368515,
0.0480896420776844,
-0.0015943425241857767,
0.01316667627543211,
0.023220950737595558,
-0.06981541961431503,
0.0036120631266385317,
0.009421114809811115,
0.0... | |
d9885c6141d52eb5b002d619714642279d4f1fca | subsection | 3 | 18 | BART Prior | BART entails both a prior on the parameter
\Theta = ((T_1,M_1),\ldots ,(T_m,M_m), \sigma )
and a MCMC algorithm for exploring the posterior.
Note that the dimension of each T_j is not fixed.The prior has the formp(\Theta ) = p(\sigma ) \, \prod _{j=1}^m \, p(T_j,M_j)with p(T,M) = p(T) \, p(M \,|\:T).
Note that the dime... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.046181876212358475,
-0.0019029420800507069,
-0.04627344757318497,
0.04102342575788498,
0.008928089402616024,
-0.007207333575934172,
0.0014756147284060717,
0.006928807590156794,
0.023792976513504982,
0.04087080806493759,
-0.027333687990903854,
-0.003353756619617343,
-0.024769723415374756,
... | |
2c09b778fd632211e9a8eca5aed069f5376d36f2 | subsection | 4 | 18 | BART MCMC | Given the observed data y, the BART model induces a
posterior distributionp((T_1,M_1), \ldots ,(T_m,M_m),\sigma | \,y)on all the unknowns that determine a sum-of-trees model
(REF and REF ). Although the sheer size of the parameter
space precludes exhaustive calculation, the following backfitting
MCMC algorithm can be u... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.034065719693899155,
-0.019001711159944534,
-0.04948076233267784,
0.01781124249100685,
0.02576296217739582,
-0.005948527716100216,
0.008394330739974976,
-0.014155893586575985,
0.018711725249886513,
0.02811337634921074,
-0.056165702641010284,
0.008379068225622177,
-0.01843700185418129,
0.... | |
4d2224654fdb80b9bf8545c0e00893c4826f7522 | subsection | 5 | 18 | Specification of the prior on | For p(\sigma ), we use the (conditionally) conjugate inverse chi-square distribution
\sigma ^2 \sim \nu \, \lambda /\chi _{\nu }^2. To guide the specification
of the hyperparameters \nu and \lambda , we recommend a data-informed approach in order to assign substantial probability to the entire region of
plausible \sigm... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.04373021796345711,
-0.02726653963327408,
-0.030699651688337326,
0.00034307281021028757,
-0.003684873692691326,
-0.030058803036808968,
0.039915651082992554,
0.018309932202100754,
0.019210169091820717,
0.04525604844093323,
-0.0357653982937336,
0.03979358449578285,
-0.0014886355493217707,
... | |
8b4365a39d02c789c189c88b7b0fdbfc2cc30473 | subsection | 6 | 18 | DPMBART: Fully Nonparametric Bayesian Additive Regression Trees | In this section we develop our fully nonparametric Dirichlet process mixture
additive regression tree (DPMBART) model.
Our base mode isY_i = f(x_i) + \epsilon _i.As in BART, we use a sum of trees to model the function f.
However, unlike BART, we do not want to make the restrictive assumption that the errors are iid nor... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-319-18968-0",
"end": 518,
"openalex_id": "https://openalex.org/W1196702536",
"raw": "Peter Muller, Fernando Quintana, Alejandro Jara, and Tim Hanson. Bayesian Nonparametric Data Analysis. Springer, 2015.",
"source_ref_id... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.02128385752439499,
-0.028042815625667572,
-0.023511413484811783,
0.019086813554167747,
-0.0004686834872700274,
-0.027569841593503952,
0.029782142490148544,
-0.009650207124650478,
0.03771590813994408,
0.02297740988433361,
-0.06572820991277695,
0.0028702691197395325,
-0.004497073125094175,
... | |
2dd5d05ac0936903fc250816aea72feaaacf00c0 | subsection | 7 | 18 | DPMBART: Fully Nonparametric Bayesian Additive Regression Trees | In Section REF we discuss the choice of prior for \alpha . | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.026685496792197227,
0.02036886103451252,
-0.02493087574839592,
0.026410860940814018,
-0.021482665091753006,
-0.002914196578785777,
0.017485179007053375,
0.031766269356012344,
0.010497210547327995,
0.021742042154073715,
0.006453953683376312,
0.004169132094830275,
-0.04504035785794258,
0.... | |
ede8dd7600b8f0f96b22316d6b628f7524fed2ca | subsection | 8 | 18 | Specification of Baseline Distribution Parameters | Section REF details the specification of the prior on \sigma in the BART
model with \epsilon _i \sim N(0,\sigma ^2). In this model, the error distribution is completely
determined by the parameter \sigma so this prior is all that is needed to describe the error
distribution.Our goal is to specify the family G_0 and ass... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.05454343929886818,
-0.006447846535593271,
-0.029942424967885017,
0.014513377100229263,
0.012552316300570965,
-0.01813027635216713,
0.030873356387019157,
0.015993710607290268,
0.025943998247385025,
0.03351353853940964,
-0.024372097104787827,
0.019106991589069366,
-0.024417880922555923,
0... | |
d9ac0b6e488b1179cba552ca50a5879cbfc6c745 | subsection | 9 | 18 | Specification of Baseline Distribution Parameters | Let e_i be the residuals from the multiple regression.Let k_s be a scaling for the \mu marginal.Given k_s we choose k_0 by solving:max |e_i| = k_s \frac{\sqrt{\lambda }}{\sqrt{k_o}}.The default we use is k_s=10.
This may seem like a very large value, but it must be remembered that the conjugate
form of our baseline dis... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.022715039551258087,
-0.017741834744811058,
-0.07237081974744797,
0.022165851667523384,
-0.011822804808616638,
-0.010915118269622326,
0.05314922705292702,
-0.001427317620255053,
0.03655153512954712,
0.04146372154355049,
-0.06919772922992706,
0.06870956718921661,
-0.028222179040312767,
0.... | |
1c79cb0885aff381a3879715981a5f17c0d70642 | subsection | 10 | 18 | Specification of the Prior on | The prior on \alpha is exactly the same as in Rossi (see Section 2.5).
The idea of the prior is to relate \alpha to the number of unique components.
The user chooses a minimum and maximum number of components I_{min} and I_{max}.
We then solve for \alpha _{min} so that the mode of the consequent distribution for I
equa... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.02450123429298401,
-0.012357409112155437,
-0.04433411359786987,
0.039482686668634415,
-0.017574982717633247,
0.0034802888985723257,
0.02721681259572506,
0.03698069229722023,
0.0032228429336100817,
-0.000560660264454782,
-0.06105475500226021,
0.03457023575901985,
-0.04567664861679077,
0.... | |
dc2a2905d8b7b4c2a3e6eeea3ce22181ca7b8114 | subsection | 11 | 18 | Computational Details | Our full parameter space consist of the \lbrace T_j,M_j\rbrace trees, j=1,2,\ldots ,m,
the \lbrace \theta _i=(\mu _i,\sigma _i)\rbrace , i=1,2,\ldots ,n and \alpha .Our Markov Monte Chain Monte Carlo (MCMC) algorithm is the obvious Gibbs sampler:\lbrace T_j,M_j\rbrace & \,|\:& \lbrace \theta _i=(\mu _i,\sigma _i)\rbrac... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/1271010",
"end": 942,
"openalex_id": "https://openalex.org/W2042217732",
"raw": "Dipak Dey, Peter Muller, and Debajyoti Sinha. Practical Nonparametric and Semiparametric Bayesian Statistics. Springer, 1998.",
"source_ref_id": ... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
0.004727596882730722,
-0.00038705056067556143,
-0.040262795984745026,
0.010981456376612186,
0.01528551522642374,
-0.035653483122587204,
0.013370056636631489,
0.0020509148016572,
0.029472121968865395,
0.039316512644290924,
-0.03754604980349541,
0.011904845014214516,
0.0003155070007778704,
0... | |
e3a41ac5102acaf31d02238952996dbd1e9fcf32 | subsection | 12 | 18 | Examples | In this section, we present examples to illustrate the inference provided
by the DPMBART model.In Section REF , we present simulated examples where the errors are drawn
from the t distribution with 20 degrees of freedom,
the t distribution with 3 degrees of freedom,
and the log of a gamma.
The t_{20} distribution gives... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1965,
"openalex_id": "",
"raw": "David Card. Using geographic variation in college proximity to estimate the return to schooling. In: Christofides, L.N., Swidinsky, R. (Eds.), Aspects of Labor Market Behavior: Essays in Honor of J... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.03780132159590721,
-0.0174667127430439,
-0.04658806696534157,
0.004934918135404587,
-0.008237576112151146,
0.0036477968096733093,
0.04603889584541321,
0.0179396104067564,
0.028724731877446175,
0.012089405208826065,
-0.019465086981654167,
0.03914374113082886,
-0.04332354664802551,
0.0112... | |
5856c3ad55b73e930251133c19414cff7296c1d7 | subsection | 13 | 18 | Simulated Examples | We present results for three simulated scenarios.
In each simulation we have 2,000 observations.
x_i \sim \text{Uniform}(-1,1).
The function f is f(x) = 10 \, x^3.
In our first simulation, \epsilon _i \sim t_{20}, the t-distribution with 20 degrees of freedom.
In our second simulation, \epsilon _i \sim t_3, the t-distr... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.0258311927318573,
-0.03841875120997429,
-0.057612866163253784,
-0.020246895030140877,
-0.007388514466583729,
-0.00904015451669693,
0.04287398234009743,
0.012885081581771374,
0.015455994755029678,
0.027646852657198906,
-0.03155281022191048,
0.02815035544335842,
-0.017347943037748337,
0.0... | |
e7527dd6b3889c8b5e2935dca2f9382a99385c06 | subsection | 14 | 18 | Simulated Examples | In the third column we plot the width of the interval for each f(x_i) from BART versus
the corresponding quantity from DPMBART.
We see that according to both BART and DPMBART there is considerably more uncertainty with
the log gamma errors which is plausible given the plots of the data in the first column of
Figure REF... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.03210711479187012,
-0.031283073127269745,
-0.05160944163799286,
0.00980457291007042,
-0.009125501848757267,
-0.019777007400989532,
0.01658765971660614,
0.003975239582359791,
0.03406039997935295,
0.014565704390406609,
-0.02847522683441639,
0.036593567579984665,
-0.05551601201295853,
0.02... | |
01a03bb04e8c865b1339db2b25b337258cc84ff7 | subsection | 15 | 18 | Card Data | In a famous paper,
Card uses instrumental variables to estimate the returns to education.
A standard specification of the first stage regression relates the treatment variable
years-of-schooling (ed76)
to measures of how close a subject lives to a two and a four year college (nearc2, nearc4),
experience and experience-... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jeconom.2008.01.007",
"end": 1219,
"openalex_id": "https://openalex.org/W1983382968",
"raw": "Timothy Conley, Christian Hansen, Robert McCulloch, and Peter Rossi. A semi-parametric bayesian approach to the instrumental variable pr... | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.052487995475530624,
-0.013984083198010921,
-0.055326007306575775,
-0.006145215127617121,
0.013190660625696182,
-0.019087931141257286,
0.007381124421954155,
0.02497757226228714,
0.02380269579589367,
0.043943438678979874,
-0.009337980300188065,
0.02093416452407837,
-0.02447405271232128,
0... | |
05841afe28f7eeb7fe06b792dc3a6c603079317d | subsection | 16 | 18 | Card Data | In the top panel of Figure REF we display the DPMBART 95% intervals and
use points (triangles) to plot the fits from BART.
In the bottom plot we again plot the DPMBART intervals, but use symbols (+) to plot the fitted values
from linear regression.
There appears to be a set of observations where (sorted index values 15... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.02636529505252838,
-0.014891509898006916,
-0.05324935168027878,
0.020140156149864197,
-0.004558236338198185,
-0.022199945524334908,
0.03649640455842018,
-0.016020579263567924,
0.036648981273174286,
0.034573934972286224,
-0.024519113823771477,
0.01754634827375412,
-0.03841887414455414,
0... | |
d1fae4315e5079072a5ce69c124b0ea065db396f | subsection | 17 | 18 | Conclusion | DPMBART is a substantial advance over BART in that the highly restrictive
and unrealistic assumption of normal errors is relaxed.
Our Bayesian ensemble modeling is now fully nonparametric.
A model with both flexible fitting of the response function f
and the error distribution has the potential to
uncover the essential... | {
"cite_spans": []
} | 1807.00068 | Fully Nonparametric Bayesian Additive Regression Trees | [
"Edward George",
"Prakash Laud",
"Brent Logan",
"Robert McCulloch",
"Rodney Sparapani"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
-0.00953883957117796,
-0.01952028088271618,
-0.03757539764046669,
0.044016022235155106,
-0.00389756984077394,
-0.05656150355935097,
0.05762985348701477,
-0.0028864527121186256,
0.04310029372572899,
0.045206468552351,
-0.05640888214111328,
0.005276885814964771,
0.004712186753749847,
0.02351... | |
04e211105420a698935045e8c56fdac7ce0a531d | abstract | 0 | 78 | Abstract | For an $n$-dimensional lattice simplex $\Delta_{(1,\mathbf{q})}$ with
vertices given by the standard basis vectors and $-\mathbf{q}$ where
$\mathbf{q}$ has positive entries, we investigate when the Ehrhart
$h^*$-polynomial for $\Delta_{(1,\mathbf{q})}$ factors as a product of
geometric series in powers of $z$. Our moti... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.034727320075035095,
0.008674200624227524,
-0.02653374709188938,
-0.03149261325597763,
0.0006050770170986652,
-0.03628363832831383,
0.04287511855363846,
0.03140106424689293,
-0.013343163765966892,
0.01400688849389553,
-0.019148850813508034,
0.029783710837364197,
-0.028334196656942368,
-0... | |
f7d652df1afe0aca75927a8951430f444f0ebef7 | subsection | 1 | 78 | Background and Motivation | Assume for this paper that P is a full-dimensional lattice polytope in \mathbb {R}^n, i.e. P is given by the convex hull of a finite subset of \mathbb {Z}^n and the affine hull of P has dimension n.
Letting tP denote the dilation of P by t, the Ehrhart polynomial L_P(t) is defined to be the degree n polynomial satisfyi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 430,
"openalex_id": "",
"raw": "Eugène Ehrhart. Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris, 254:616–618, 1962.",
"source_ref_id": "54a82ff22db5e99a9629e0b6ba5f94edb734beec",
"star... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.027528656646609306,
-0.002512142527848482,
0.00903379451483488,
0.010681851767003536,
0.006859274581074715,
-0.04953327029943466,
-0.00025703205028548837,
0.0003051957464776933,
-0.02736079879105091,
0.03961440920829773,
-0.03958388790488243,
0.044161826372146606,
-0.051639121025800705,
... | |
51c1725f6185ea6d2b7aa59b6cf051a7a3f5ab75 | subsection | 2 | 78 | Background and Motivation | Our connection to Ehrhart positivity is provided by the following theorem, which is a special case of a more general result proved by Rodriguez-Villegas.Theorem 1.1 (Rodriguez-Villegas )
If f(t)\in \mathbb {R}[t] is of degree n and the associated polynomial \sum _{j=0}^nh_j^*z^j is also of degree n with all roots on ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0002-9939-02-06454-7",
"end": 393,
"openalex_id": "https://openalex.org/W2150346305",
"raw": "Fernando Rodriguez-Villegas. On the zeros of certain polynomials. Proc. Amer. Math. Soc., 130(8):2251–2254 (electronic), 2002.",
"s... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
0.000908699119463563,
-0.010130135342478752,
-0.019939890131354332,
-0.012250751256942749,
-0.0027403999119997025,
-0.05602087080478668,
0.017285306006669998,
0.02364715375006199,
-0.028956321999430656,
0.016675056889653206,
-0.04482279717922211,
0.0209162887185812,
-0.07200939953327179,
0... | |
4cba77e10b116d05d8a06e482861c1507438569c | subsection | 3 | 78 | Our Contributions | One way for an h^*-polynomial to be Kronecker is to factor as a product of geometric series in powers of z, which we refer to as a geometric factorization.
Motivated by Corollary REF , we explore geometric factorizations for lattice simplices of the following form: let \Delta _{(1,{{q}})} be the simplex with vertices g... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00454-007-9002-5",
"end": 625,
"openalex_id": "https://openalex.org/W2164000301",
"raw": "Sam Payne. Ehrhart series and lattice triangulations. Discrete Comput. Geom., 40(3):365–376, 2008.",
"source_ref_id": "c4a929995eec8d23... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03695720061659813,
0.03012118674814701,
-0.027389833703637123,
-0.014091650024056435,
-0.0066910539753735065,
-0.004337359219789505,
0.06329416483640671,
0.03036533109843731,
-0.011078004725277424,
0.03228795900940895,
-0.015045334585011005,
0.018295492976903915,
-0.0212404727935791,
-0... | |
0c89b245b034531cd58419123eb13a7534251653 | subsection | 4 | 78 | Definition and Reflexivity | Given a vector of positive integers {{q}}\in \mathbb {Z}_{>0}^n, we define\Delta _{(1,{{q}})}:= \mathrm {conv}\left\lbrace {{e}}_1,\ldots ,{{e}}_n,-\sum _{i=1}^n q_i{{e}}_i\right\rbracewhere {{e}}_i denotes the i-th standard basis vector in \mathbb {R}^n.
There is a natural stratification of the family of simplices of ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.05244722589850426,
0.013676246628165245,
-0.036764923483133316,
-0.02439299412071705,
-0.005995275918394327,
0.008481866680085659,
0.017162049189209938,
0.00986245833337307,
-0.007513164076954126,
0.0070936474949121475,
-0.003659330541267991,
0.001616093097254634,
-0.008718321099877357,
... | |
faae9ec91da2f7b0fabd82e0d3b4f5873057f5be | subsection | 5 | 78 | Definition and Reflexivity | It is straightforward to show that \Delta _{(1, {{q}})} is reflexive if and only ifq_i \text{ divides } 1 + \sum _{j =1}^n q_j, \quad \text{ for all $1 \le i \le n$ } \, .Equivalently, if {{q}} is supported by {{r}} with multiplicity {{x}}, then \Delta _{(1,{{q}})} is reflexive if and only if if \mathrm {lcm}\left(r_1... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s002290100235",
"end": 172,
"openalex_id": "https://openalex.org/W2050266449",
"raw": "Heinke Conrads. Weighted projective spaces and reflexive simplices. Manuscripta Math., 107(2):215–227, 2002.",
"source_ref_id": "2871781dca... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.050836578011512756,
0.010985094122588634,
-0.0025670030154287815,
-0.013624568469822407,
-0.016706498339772224,
0.006167673040181398,
0.0070525831542909145,
0.022397387772798538,
0.0009359262185171247,
0.002633752766996622,
-0.0018461061408743262,
-0.0183237474411726,
-0.00984081346541643... | |
95c119326dd1e1f35e0fe30f1d3b21084b0b6003 | subsection | 6 | 78 | Definition and Reflexivity | By the definition of s_i, we can verify that\mathrm {lcm}\left(s_1,\dots ,s_d\right) = \frac{\mathrm {lcm}\left(r_1,\dots ,r_d\right)}{\gcd (r_1,\dots ,r_d)} = \mathrm {lcm}\left(r_1,\dots ,r_d\right).Our analysis of families of {{q}}-vectors will require a precise language for studying R-multiplicities of vectors, whi... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04251200333237648,
-0.0023403726518154144,
-0.00920889899134636,
0.019501198083162308,
-0.004230600316077471,
0.026047375053167343,
-0.01273376401513815,
-0.008323868736624718,
-0.007618132513016462,
0.000041247400076827034,
-0.03555382788181305,
0.0024242980871349573,
0.00955985952168703... | |
f5db60559c29a41ed96779dc9d2d9749bbea62b4 | subsection | 7 | 78 | Definition and Reflexivity | Plugging in x_i = c_i s_i + \rho _i and using the fact that s_i r_i = \mathrm {lcm}\left(r_1, \dots , r_m\right), we obtain1 + \sum _{i=1}^dx_i r_i = 1+ \sum _{i=1}^d (c_i s_i r_i + \rho _i r_i) \equiv 1+ \sum _{i=1}^d \rho _i r_i \mathrm {}c, ) be the (unique) s-division of x such that i si for each i.
As 0 i < si, we... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
0.004440978169441223,
0.0192442387342453,
-0.06690463423728943,
0.019641028717160225,
-0.033269185572862625,
0.020465126261115074,
0.017122946679592133,
0.019808899611234665,
-0.01010284386575222,
-0.0019324360182508826,
-0.024295661598443985,
-0.013712093234062195,
-0.004303628578782082,
... | |
ec5f768ecae66b431ffe68dd0fcf58b190825a23 | subsection | 8 | 78 | Body | The following theorem shows that the h^*-polynomial for any \Delta _{(1,{{q}})} can be expressed purely in terms of the vector {{q}}.Theorem 2.11 (Braun, Davis, and Solus )
The h^*-polynomial of \Delta _{(1,{{q}})} is given by\sum _{b=0}^{q_1+q_2+\cdots +q_n}z^{w(b)}wherew(b)=b-\sum _{i=1}^n\left\lfloor \frac{bq_i}{1+... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.aam.2018.06.003",
"end": 358,
"openalex_id": "https://openalex.org/W2963717057",
"raw": "Benjamin Braun, Robert Davis, and Liam Solus. Detecting the integer decomposition property and ehrhart unimodality in reflexive simplices. to... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.021940836682915688,
0.03439126908779144,
-0.020140405744314194,
-0.038571927696466446,
-0.019758958369493484,
0.016051292419433594,
0.04241691529750824,
0.02726583741605282,
-0.009124213829636574,
0.025511180981993675,
-0.005744593217968941,
0.030332671478390694,
-0.011672280728816986,
... | |
958412863af15d905d479eeb334c081780bf2173 | subsection | 9 | 78 | Body | We say a polynomial f(z) in z is a product of geometric series (in powers of z) if there exists p \in \mathbb {Z}_{>0}, e_1, e_2, \dots , e_p \in \mathbb {Z}_{>0} and \gamma _1, \dots , \gamma _p \in \mathbb {Z}_{\ge 2} such thatf(z) = \prod _{j=1}^p \sum _{i=0}^{\gamma _j-1} z^{i e_j} = \prod _{j=1}^p \left( 1 + z^{e_... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.033226292580366135,
0.009168503805994987,
-0.027734342962503433,
-0.01653686910867691,
0.008329455740749836,
-0.009839741513133049,
0.04793250933289528,
0.036063797771930695,
-0.029595503583550453,
-0.001842091209255159,
-0.03203636780381203,
0.010564373806118965,
-0.012433161959052086,
... | |
12b672b63d308f94b31eeb8091cc8081381cec34 | subsection | 10 | 78 | Body | Hence \Delta _{(1,{{q}})} is supported on one integer r=1.A natural next step is to consider {{q}} that are supported by more than two integers.
Experimental computation and Proposition REF suggest that a starting point for such an exploration are 3-supported {{q}}'s with {{s}} entries coprime.
When \gcd (a,b)=1 and {{... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03546545282006264,
0.011063878424465656,
-0.030795732513070107,
-0.016801834106445312,
-0.01962503045797348,
0.026827996596693993,
0.050268158316612244,
0.015947245061397552,
-0.005848594941198826,
0.021166343241930008,
-0.021059520542621613,
-0.0077218241058290005,
0.008355136029422283,
... | |
a3a952aa9e2549a19898662ba3d80b92511bcc0c | subsection | 11 | 78 | Body | In this case, if we fix all elements except for the following pairs which are exchanged by the bijection, then the factorization follows:& (2,2,0)\longleftrightarrow (0,0,1), \, \, \, (4,1,0)\longleftrightarrow (1,0,1)\\
& (3,2,0)\longleftrightarrow (0,1,1), \, \, \, (4,2,0)\longleftrightarrow (2,0,1)Similarly for case... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.0332690104842186,
0.004894970450550318,
-0.040807951241731644,
-0.003998385742306709,
-0.018679480999708176,
-0.017305990681052208,
0.05185692384839058,
0.03189551830291748,
-0.012101983651518822,
0.017168641090393066,
0.0016205288702622056,
-0.01112527959048748,
-0.003948787227272987,
... | |
edcffd5375cd9019c141a067f9d2630803d8f11b | subsection | 12 | 78 | Body | For {{q}}=({{r}},{{x}}), the following three cases imply that h^*(\Delta _{(1,{{q}})};z) is Kronecker.(c_1,c_2,c_3)=(1,3,1), where
g_{(6,10,15)}^{(4,8,3)}(z)=(1+z^3)(1+z^2+z^4)(1+z+z^2+z^3+z^4)^2
(c_1,c_2,c_3)=(c,c,4c-1) for c\ge 1, where
g_{(6,10,15)}^{(5c-1,3c-1,2(4c-1)+1)}(z)=(1+z^{4c-1})(1+z^c+z^{2c})(1+z+z^c+z... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.013244165107607841,
0.021956628188490868,
-0.03301886096596718,
-0.031370971351861954,
-0.02279583178460598,
-0.005538746248930693,
0.05325130745768547,
0.047971952706575394,
-0.02824302949011326,
0.011382658034563065,
-0.012862708419561386,
0.00319469696842134,
-0.00451262854039669,
0.... | |
784c655402a56d69a6656d178c7270200dd833da | subsection | 13 | 78 | Body | A search over (pairwise coprime) {{s}} and {{x}} with 2\le s_i\le 11 and 1\le x_i\le 50 has produced only two further examples of 3-supported {{q}}'s with Kronecker h^*-polynomials, specifically:{{s}}=(11,4,3), \, {{r}}=(12, 33, 44), \, {{x}}=(21, 11, 22)and{{s}}=(10,7,3), \, {{r}}=(21, 30, 70), \, {{x}}=(9, 10, 5)For ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03988555818796158,
-0.0037669269368052483,
-0.04745374247431755,
-0.03445355221629143,
-0.05447262525558472,
-0.016387563198804855,
0.054167453199625015,
0.03037955053150654,
-0.018432192504405975,
-0.002719816518947482,
-0.004150295164436102,
-0.0067480443976819515,
0.010802973993122578,... | |
3681518cde0268569e2344fbfad10b9613d22f0d | subsection | 14 | 78 | Free Sums Create New Kronecker | For two reflexive simplices \Delta _{(1,{{q}})} and \Delta _{(1,{{p}})} with Kronecker h^*-polynomials, there exists an operation that produces a new simplex \Delta _{(1,{{y}})} that is reflexive with a Kronecker h^*-polynomial.
We say that P\oplus Q:=\mathrm {conv}\left\lbrace P\cup Q\right\rbrace is an affine free su... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00026-016-0337-6",
"end": 1341,
"openalex_id": "https://openalex.org/W1486480825",
"raw": "Benjamin Braun and Robert Davis. Ehrhart series, unimodality, and integrally closed reflexive polytopes. Ann. Comb., 20(4):705–717, 2016.",
... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.005364434793591499,
0.0005374926840886474,
-0.04645615816116333,
-0.0334533154964447,
0.015185242518782616,
0.014704503118991852,
0.02556309662759304,
0.055216286331415176,
0.02194610796868801,
0.018848014995455742,
-0.014277179725468159,
-0.0030179715249687433,
-0.015475211665034294,
0... | |
faf8bed1cc192a88381aeb367ebf6ee3bad1b796 | subsection | 15 | 78 | Free Sums Create New Kronecker | Thus, the resulting h^*-polynomials are also Kronecker when the summands have Kronecker h^*-polynomials. | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.040436964482069016,
0.020371073856949806,
-0.04821917414665222,
-0.03662215545773506,
-0.05575723201036453,
-0.020584704354405403,
0.0055467309430241585,
0.038697414100170135,
-0.006931506097316742,
0.004066585097461939,
-0.06189144402742386,
0.010956128127872944,
-0.03946037217974663,
... | |
07307f026962c4ec073c4567351388871c68df43 | subsection | 16 | 78 | Reflexive | In this subsection, we show that for a reflexive \Delta _{(1,{{q}})}, it is always possible to factor a geometric series from h^*(\Delta _{(1,{{q}})};z).
The following polynomial plays a fundamental role in this factorization.Definition 3.1
Suppose {{r}}, {{x}}, \ell and {{s}} are as given in Setup REF .
We defineg_{{... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04451253265142441,
0.019434191286563873,
-0.011273050680756569,
-0.017740944400429726,
-0.009076407179236412,
0.0026371160056442022,
0.05735679343342781,
0.02565801329910755,
0.005228468216955662,
0.002646649954840541,
0.001748543232679367,
0.03154623508453369,
-0.027564821764826775,
0.... | |
a0f93d53eb26b2c6f8cb9687d7f153b8ec4b2b71 | subsection | 17 | 78 | Reflexive | Then using (REF ) we have:w(b)=w(\alpha \ell +\beta ) & = \alpha \ell +\beta - \sum _{i=1}^d x_i\left\lfloor \frac{(\alpha \ell +\beta )r_i}{\ell M} \right\rfloor \\
& = \beta + \alpha \ell - \sum _{i=1}^d x_i\left\lfloor \frac{\alpha \ell +\beta }{\ell s_i} \right\rfloor = \beta + \alpha \ell - \sum _{i=1}^d x_i\left\... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.026996739208698273,
0.039587076753377914,
-0.026615213602781296,
-0.054359741508960724,
-0.03604651987552643,
0.02124333567917347,
0.014352986589074135,
0.03552764654159546,
-0.010209619998931885,
0.015169451013207436,
-0.01942727342247963,
0.030964603647589684,
-0.017840128391981125,
0... | |
42faa41881d4bbcedcfedf26c8ea8df570d0a18f | subsection | 18 | 78 | Reflexive | In this case,h^*(\Delta _{(1,{{q}})}; z) =& 1 + z + 2 z^2 + 4z^3 + 4z^4 + 5z^5 + 6 z^6 + 5z^7 + 4z^8 + 4 z^9 + 2z^{10} + z^{11} + z^{12},
{which can be factored as (1 + z^2) (1+z^3)^2 (1+z+ z^2 +z^3+z^4), and}
g_{{r}}^{{x}}(z) =& 1 + z^2 + 2z^3 + z^4 + z^5 + 2z^6 + z^7 + z^9,which cannot be written as a product of geom... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/tran/7424",
"end": 527,
"openalex_id": "https://openalex.org/W2962924579",
"raw": "Liam Solus. Simplices for numeral systems. to appear in Transactions of the American Mathematical Society, preprint at https://arxiv.org/abs/1706.004... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.037636421620845795,
0.014178521931171417,
-0.013865647837519646,
-0.009431998245418072,
-0.018375609070062637,
-0.004715999122709036,
0.054882630705833435,
0.040078360587358475,
-0.004780863411724567,
0.023824188858270645,
-0.02319844253361225,
0.007108338642865419,
-0.03391246125102043,
... | |
057e9986bc280f57741f0d1f56c32299844d23db | subsection | 19 | 78 | Reflexive | Further,\mathrm {lcm}\left(r_1,\ldots ,r_d,M\right)=\mathrm {lcm}\left(r_1,\ldots ,r_d\right)and thusg_{({{r}},M)}^{({{x}},y)}(z):=\sum _{0\le \alpha < \mathrm {lcm}\left(r_1,\ldots ,r_d\right)}z^{u(\alpha )}whereu_{({{r}},M)}^{({{x}},y)}(\alpha ) = \alpha (\ell + y) - \sum _{i=1}^d x_i\left\lfloor \frac{\alpha }{s_i} ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.0347275584936142,
0.011268147267401218,
-0.03387310355901718,
-0.04250921681523323,
0.008903132751584053,
0.014960620552301407,
0.016494065523147583,
0.026335574686527252,
0.007201849017292261,
0.007667222525924444,
-0.012053942307829857,
0.02450459636747837,
-0.02397056110203266,
0.050... | |
5e196e7b987821c4f85df1747726147a07fce4f6 | subsection | 20 | 78 | Reflexive | Then using (REF ) we have:w(b)=w(\alpha \ell +\beta ) & = \alpha \ell +\beta - \sum _{i=1}^d x_i\left\lfloor \frac{(\alpha \ell +\beta )r_i}{\ell M} \right\rfloor \\
& = \beta + \alpha \ell - \sum _{i=1}^d x_i\left\lfloor \frac{\alpha \ell +\beta }{\ell s_i} \right\rfloor = \beta + \alpha \ell - \sum _{i=1}^d x_i\left\... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.026996739208698273,
0.039587076753377914,
-0.026615213602781296,
-0.054359741508960724,
-0.03604651987552643,
0.02124333567917347,
0.014352986589074135,
0.03552764654159546,
-0.010209619998931885,
0.015169451013207436,
-0.01942727342247963,
0.030964603647589684,
-0.017840128391981125,
0... | |
c9eb4df69c2790ec48169ab9bf0b9bf8f0c194e5 | subsection | 21 | 78 | Reflexive | In this case,h^*(\Delta _{(1,{{q}})}; z) =& 1 + z + 2 z^2 + 4z^3 + 4z^4 + 5z^5 + 6 z^6 + 5z^7 + 4z^8 + 4 z^9 + 2z^{10} + z^{11} + z^{12},
{which can be factored as (1 + z^2) (1+z^3)^2 (1+z+ z^2 +z^3+z^4), and}
g_{{r}}^{{x}}(z) =& 1 + z^2 + 2z^3 + z^4 + z^5 + 2z^6 + z^7 + z^9,which cannot be written as a product of geom... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/tran/7424",
"end": 527,
"openalex_id": "https://openalex.org/W2962924579",
"raw": "Liam Solus. Simplices for numeral systems. to appear in Transactions of the American Mathematical Society, preprint at https://arxiv.org/abs/1706.004... | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.037636421620845795,
0.014178521931171417,
-0.013865647837519646,
-0.009431998245418072,
-0.018375609070062637,
-0.004715999122709036,
0.054882630705833435,
0.040078360587358475,
-0.004780863411724567,
0.023824188858270645,
-0.02319844253361225,
0.007108338642865419,
-0.03391246125102043,
... | |
af1da0d097b275bb568f8ea90fec0633a9d11af6 | subsection | 22 | 78 | Reflexive | Further,\mathrm {lcm}\left(r_1,\ldots ,r_d,M\right)=\mathrm {lcm}\left(r_1,\ldots ,r_d\right)and thusg_{({{r}},M)}^{({{x}},y)}(z):=\sum _{0\le \alpha < \mathrm {lcm}\left(r_1,\ldots ,r_d\right)}z^{u(\alpha )}whereu_{({{r}},M)}^{({{x}},y)}(\alpha ) = \alpha (\ell + y) - \sum _{i=1}^d x_i\left\lfloor \frac{\alpha }{s_i} ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.02468474581837654,
0.022304756566882133,
-0.02453218214213848,
-0.04305336996912956,
0.021999631077051163,
0.016049660742282867,
-0.05007128044962883,
0.0012853459920734167,
0.006880605593323708,
-0.015134281478822231,
-0.026744350790977478,
0.03066522814333439,
-0.018551699817180634,
0... | |
74d422fb178512f3e1fc917dee1d042cd757380f | subsection | 23 | 78 | A Useful Form for | Our goal in this subsection is to prove Theorem REF below, providing a reformulation of g_{{r}}^{{x}}(z) that is helpful for establishing factorizations.
We will require the following theorem from elementary number theory.Theorem 3.7 (Generalized Chinese Remainder Theorem)
Suppose m_1, m_2, \dots , m_d are positive in... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04247022047638893,
0.005026558414101601,
-0.02631506137549877,
-0.022318223491311073,
0.004942655097693205,
0.015804292634129524,
0.003081531962379813,
-0.012272734194993973,
-0.0035754160489887,
0.02138766273856163,
-0.05943390354514122,
-0.022333478555083275,
-0.03395787253975868,
0.0... | |
f35dea013ab47913a95b2e12fa87e43ec0f27da4 | subsection | 24 | 78 | A Useful Form for | Theng_{{r}}^{{x}}(z) = \sum _{{{i}}\in I({{r}})} z^{\sum _{j=1}^d (c_j i_j - \rho _j \omega _j({{i}}))} \, .By Definition REF and Corollary REF , it is enough to verify that for each {{i}}\in I({{r}}), we haveu(\alpha ({{i}}))= \alpha ({{i}}) \ell - \sum _{j=1}^d x_j\left\lfloor \frac{\alpha ({{i}})}{s_j} \right\rfloor... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04612201079726219,
0.017932945862412453,
-0.04313063994050026,
0.010660563595592976,
-0.017261413857340813,
-0.005681309849023819,
0.010408739559352398,
-0.0005990366917103529,
-0.02200792171061039,
0.012240189127624035,
-0.0347975455224514,
0.008562027476727962,
0.0007020557532086968,
... | |
71a783a9f41a86246b7349f7b6cbd6b694f4a453 | subsection | 25 | 78 | A Useful Form for | Hence,
\displaystyle -\sum _{t=1}^d \rho _{t} r_{t} i_{t} \equiv -\rho _j r_j i_j \pmod {s_j}. Next, it follows from Lemma REF that \displaystyle \sum _{j=1}^d \rho _t r_t \equiv -1 \pmod {s_j}. Again, as s_j divides r_t whenever t \ne j, we conclude that \rho _j r_j \equiv -1 \pmod {s_j}. Thus, (REF ) follows.By the d... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.029460720717906952,
0.029552262276411057,
-0.042200081050395966,
-0.012472366914153099,
-0.003148605115711689,
-0.004458775743842125,
0.0059234206564724445,
0.019192950800061226,
-0.013792072422802448,
0.007788554299622774,
-0.048424821346998215,
-0.010122831910848618,
-0.0462583675980567... | |
00c78b743a6d7c14cb5edf8d98522d44c104816a | subsection | 26 | 78 | A Useful Form for | Moreover, when there is a solution, it is unique modulo \mathrm {lcm}\left(m_1, m_2, \dots , m_d\right).Motivated by the above theorem, for two vectors {{r}} and {{s}} related by (REF ) we defineI = I({{r}}) := \lbrace {{i}}= (i_1, \dots , i_d) \in \left\langle s_1 \right\rangle \times \cdots \times \left\langle s_d \r... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.02970281057059765,
0.0014321270864456892,
-0.027780594304203987,
0.016094744205474854,
-0.010190795175731182,
-0.005667485296726227,
-0.007692676968872547,
0.012120638974010944,
-0.004092336632311344,
0.004797911737114191,
-0.04408891871571541,
-0.00004981933307135478,
-0.0093746157363057... | |
7af146cee55722e65f440b51172b93e8a4d14fe6 | subsection | 27 | 78 | A Useful Form for | Recall that (a \bmod b) is the unique integer a^{\prime } \in \left\langle b \right\rangle satisfying a \equiv a^{\prime } \pmod {b}.Proposition 3.11
Assume Setup REF where s_1, \dots , s_d are pairwise coprime. Then\mathrm {lcm}\left(r_1, \dots , r_d\right) = \mathrm {lcm}\left(s_1, \dots , s_d\right) = s_1 s_2 \dots... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.015451199375092983,
0.016130289062857628,
-0.039249859750270844,
0.0075157685205340385,
-0.019533367827534676,
-0.005440348293632269,
0.027804527431726456,
0.02170035056769848,
-0.0014926621224731207,
-0.007771380711346865,
-0.03723548352718353,
-0.012246506288647652,
-0.00713807251304388... | |
12f4180a51da5144b2313297f47a7c555f088354 | subsection | 28 | 78 | A Useful Form for | Hence, (REF ) is equivalent to\omega _j({{i}}) \equiv \sum _{t \ne j} \rho _{t} \frac{r_{t}}{s_j} (i_j-i_t) \pmod {r_j},By (REF ), we have that \alpha ({{i}}) = -\sum _{t=1}^d \rho _t r_t i_t + M s_1 s_2 \dots s_d = -\sum _{t=1}^d \rho _t r_t i_t + M s_j r_j for some integer M. Hence,\omega _j({{i}}) = \frac{\alpha ({{... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.020116806030273438,
0.03199152275919914,
-0.04017255827784538,
-0.011119190603494644,
-0.03208310157060623,
-0.011248927563428879,
0.018514176830649376,
0.04240097478032112,
0.007589591667056084,
-0.017476284876465797,
-0.02715310826897621,
0.013393396511673927,
-0.06157146766781807,
0.... | |
37319bd8e9530ff1154d25d4544f249296886d1e | subsection | 29 | 78 | Some Kronecker | We have seen in Proposition REF that any reflexive \Delta _{(1,{{q}})} supported on one integer has {{r}}=(1).
The next level of complexity of {{q}}-vectors are those for which {{q}} has two distinct entries.
Payne's simplices from Example REF are an important example of this type in Ehrhart theory, as they are reflexi... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.0061160423792898655,
0.01571933552622795,
-0.017901727929711342,
-0.006077888887375593,
-0.01861901953816414,
0.007230131421238184,
0.057810526341199875,
0.028264280408620834,
0.01040833629667759,
0.016482409089803696,
-0.013742972165346146,
0.025624042376875877,
-0.01900055631995201,
0... | |
0db69f717ab06f2d6aa4b5091dd8bfa9727a901a | subsection | 30 | 78 | Setup | Recall from elementary number theory that for {{r}}=(r_1,r_2) \in \left( \mathbb {Z}_{>0} \right)^2 such that \gcd (r_1,r_2) = 1, there exists an integer solution {{\rho }}= (\rho _1, \rho _2) to \rho _1 r_1 + \rho _2 r_2 = -1. Furthermore, if {{\rho }}^* = (\rho _1^*, \rho _2^*) is a special integer solution to \rho _... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.012951641343533993,
0.011708344332873821,
-0.02013682760298252,
0.02327939309179783,
0.003386648138985038,
-0.009450579062104225,
0.049304716289043427,
0.02420995756983757,
-0.0031158688943833113,
0.009061572141945362,
-0.008474248461425304,
-0.019663916900753975,
-0.018550289794802666,
... | |
a45664cf21c421402c7a1d442fa5869046c08c2f | subsection | 31 | 78 | Setup | Then for each {{i}}= (i_1, i_2) \in I({{r}}) = \left\langle r_2 \right\rangle \times \left\langle r_1 \right\rangle , we have:\alpha ({{i}}) =& \left( -\rho _1 r_1 i_1 - \rho _2 r_2 i_2 \bmod r_1 r_2 \right) \\
\omega _1({{i}}) =& \left( \rho _2 (i_1 - i_2) \bmod r_1 \right) \\
\omega _2({{i}}) =& \left( \rho _1 (i_2 -... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.00040952954441308975,
0.037255268543958664,
-0.02750665508210659,
0.0012014138046652079,
-0.01179292518645525,
-0.00543115334585309,
0.022273831069469452,
0.019497230648994446,
-0.01845981925725937,
-0.007197040598839521,
-0.024836847558617592,
-0.00936721358448267,
-0.035119421780109406,... | |
84210375595b5c5c20bd5e20b5bcec7791eac698 | subsection | 32 | 78 | Setup | The proof is complete after we show that the right hand side of (REF ) is equal to M + k \omega _1({{i}}), which is equivalent to0 \le M + k \omega _1({{i}}) \le ka-2 \, .It is straightforward to verify the left-hand inequality0 \le M + k \omega _1({{i}})=\left\lfloor \frac{i_1-i_2}{a} \right\rfloor + k \left( (i_1 - i... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.013819604180753231,
0.024113722145557404,
-0.015643395483493805,
-0.01655910722911358,
-0.00827955361455679,
-0.0038421705830842257,
0.06428290903568268,
0.040413375943899155,
0.027074521407485008,
-0.0028768586926162243,
-0.028341254219412804,
-0.01726115122437477,
-0.026876116171479225,... | |
bdc412e2a9719fe2e849df777af67f7ae04684c0 | subsection | 33 | 78 | Setup | Let {{q}} be the vector supported by {{r}} with the R-multiplicity {{x}}= (x_1, x_2) \in \left( \mathbb {Z}_{>0} \right)^2 having the property that {{\rho }} is an {{s}}-remainder of {{x}}; that is, for some integers c_1, c_2,x_1 = c_1 s_1 + \rho _1 \text{ and } x_2 = c_2 s_2 + \rho _2 \, .Thus, \ell =\ell ({{q}})= c_1... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03417310118675232,
0.01557622104883194,
-0.001747748116031289,
0.03878036513924599,
-0.014401520602405071,
-0.014485428109765053,
0.047811828553676605,
0.040397487580776215,
-0.018001900985836983,
-0.0033906123135238886,
-0.006617224309593439,
-0.007208388298749924,
-0.0021091210655868053... | |
b94b0e76806ea62178184e89bf06d9c5cc9d5a5d | subsection | 34 | 78 | Setup | Hence, we only need to show that, using the notation from Definition REF ,-k \omega _1({{i}}) + \omega _2({{i}}) = \left\lfloor \frac{i_1-i_2}{a} \right\rfloor = M \, ,and the result follows from Theorem REF .
Applying Corollary REF , we get\omega _1({{i}}) = \left( (i_1 - i_2) \bmod a \right) \quad \text{and} \quad \o... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.043362848460674286,
0.017760151997208595,
-0.039365287870168686,
-0.039334770292043686,
-0.02149832993745804,
0.013266712427139282,
0.034909989684820175,
0.06261827051639557,
0.021406782791018486,
-0.016478491947054863,
-0.042477890849113464,
-0.009559051133692265,
-0.014655177481472492,
... | |
dc130802d36a682785103f76a51779e631379474 | subsection | 35 | 78 | Four Main Theorems | Theorem 4.5
For {{r}}=(1,a) or (a,1) and any R-multiplicity {{x}}, the resulting g_{{r}}^{{x}}(z) is a geometric series, which is a Kronecker polynomial.Suppose {{r}}= (1, a) for some integer a \ge 2.
Then {{s}}=(a, 1), {{\rho }}= (-1, 0), and {{x}}= (ac_1 -1, c_2) for some positive integers c_1, c_2.
Then \omega _1({... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.013102165423333645,
0.03879522904753685,
-0.020267531275749207,
-0.02977556735277176,
-0.01418574620038271,
-0.022312598302960396,
0.04093186557292938,
0.01680312678217888,
-0.04056558758020401,
-0.008134483359754086,
-0.013216628693044186,
0.0023083314299583435,
-0.04117605462670326,
-... | |
2cf4886a7cc27353ad852836fdf4f8e0c4a16f3f | subsection | 36 | 78 | Four Main Theorems | Thus, by Lemma REF ,g_{{r}}^{{x}}(z) = \sum _{{{i}}\in \left\langle ka-1 \right\rangle \times \left\langle a \right\rangle } z^{c i_1 + ( (ka-1)c-k) i_2 - \left\lfloor \frac{i_1-i_2}{a} \right\rfloor }.One sees that it is enough to show that there exists a bijection \varphi on \left\langle ka-1 \right\rangle \times \le... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03969903290271759,
0.006636848673224449,
-0.01615729369223118,
-0.007285276427865028,
0.009367873892188072,
0.00953570194542408,
0.051538560539484024,
0.012701555155217648,
-0.007292904891073704,
0.019407059997320175,
-0.026135452091693878,
-0.040034692734479904,
-0.03081938810646534,
0... | |
e262b1799180f106b6a1a7f43f597541ada5a087 | subsection | 37 | 78 | Four Main Theorems | For {{r}}= (a, a-1) and {{x}}= ( (a-1)c-1, ac+1), we haveg_{{r}}^{{x}}(z) = (1+z^{c+1})\left(\sum _{j=0}^{2\left\lfloor \frac{a-1}{2}\right\rfloor }z^{cj}\right)\left(\sum _{j=0}^{\left\lceil \frac{a-1}{2}\right\rceil -1}z^{2cj}\right),which is a Kronecker polynomial.With the given {{x}}, we have the desirable {{s}}-di... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.017255960032343864,
0.037807490676641464,
-0.030072057619690895,
-0.023328348994255066,
0.014227394014596939,
0.005458284635096788,
0.050654102116823196,
-0.008536436595022678,
-0.042933929711580276,
-0.002172252396121621,
-0.04204900562763214,
0.00786511693149805,
-0.05492613464593887,
... | |
01c1da5d3b4a9973db0544ee9f6969a1837e89a7 | subsection | 38 | 78 | Four Main Theorems | We define a bijection \phi :\left\langle a-1 \right\rangle \times \left\langle a \right\rangle \rightarrow \left\langle a-1 \right\rangle \times \left\langle a \right\rangle by sending (i_1,i_2)\in A to the element (i_2,i_1+1)\in B and sending the element (i_1,i_2)\in B to the element (i_2-1,i_1)\in A.Now, using (REF )... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.028351908549666405,
0.02090533636510372,
-0.02668863721191883,
-0.012962833978235722,
0.009712588973343372,
-0.005020332522690296,
0.05163770541548729,
-0.009476068429648876,
-0.0036279151681810617,
0.024628622457385063,
-0.03613418713212013,
0.0019722734577953815,
-0.031586892902851105,
... | |
03ca4b2a9944513b83c1635350f8ec5b7017373d | subsection | 39 | 78 | Four Main Theorems | For convenience, for any {{i}}\in \mathbb {Z}^2, we letu({{i}}) := c i_1 + (ac-1) i_2 - \left\lfloor \frac{i_1-i_2}{a} \right\rfloor .It is straightforward to verify that for any m=1,2,\dots , a-1, we haveu( ma-1, a-1) = u (a^2-1, m-1).Notice thatI^{\prime } := \left\langle a^2 \right\rangle \times \left\langle a \righ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.020616192370653152,
0.013001477345824242,
-0.0222795270383358,
-0.0024854643270373344,
0.026918550953269005,
0.016679124906659126,
0.022722065448760986,
0.015290469862520695,
-0.006508368533104658,
0.022126926109194756,
-0.0509987510740757,
-0.005974270403385162,
-0.018083039671182632,
... | |
a6227f5e076d01095eb6cf76f12f5b4411f89c79 | subsection | 40 | 78 | Four Main Theorems | Sincea i_2 + i_1 = (a+1) i_2 + (i_1-i_2) = (a+1)(i_2-1) + (a+1+i_1-i_2)and -(a-1) \le i_1 - i_2 \le a-1, we conclude that if (j_2,j_3) = \Psi (i_1,i_2), thenj_2= i_1-i_2 - (a+1) \left\lfloor \frac{i_1-i_2}{a}\right\rfloor , \quad j_3 = i_2 + \left\lfloor \frac{i_1-i_2}{a}\right\rfloor .Using the above, it is easy to ve... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.011247079819440842,
0.031528446823358536,
-0.03931136801838875,
-0.019579380750656128,
-0.0008135819807648659,
-0.017854928970336914,
0.04068482294678688,
0.020037198439240456,
-0.03986074775457382,
0.005215317942202091,
-0.01870952546596527,
0.006310267839580774,
-0.03525204211473465,
... | |
2fe6c66ea9047e12032ac1a3f7b809ea334ebebd | subsection | 41 | 78 | Four Main Theorems | Thus, by Lemma REF ,g_{{r}}^{{x}}(z) = \sum _{{{i}}\in \left\langle ka-1 \right\rangle \times \left\langle a \right\rangle } z^{c i_1 + ( (ka-1)c-k) i_2 - \left\lfloor \frac{i_1-i_2}{a} \right\rfloor }.One sees that it is enough to show that there exists a bijection \varphi on \left\langle ka-1 \right\rangle \times \le... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03969903290271759,
0.006636848673224449,
-0.01615729369223118,
-0.007285276427865028,
0.009367873892188072,
0.00953570194542408,
0.051538560539484024,
0.012701555155217648,
-0.007292904891073704,
0.019407059997320175,
-0.026135452091693878,
-0.040034692734479904,
-0.03081938810646534,
0... | |
2885539acb5603641a69f91b6e206b5cbc937dfd | subsection | 42 | 78 | Four Main Theorems | For {{r}}= (a, a-1) and {{x}}= ( (a-1)c-1, ac+1), we haveg_{{r}}^{{x}}(z) = (1+z^{c+1})\left(\sum _{j=0}^{2\left\lfloor \frac{a-1}{2}\right\rfloor }z^{cj}\right)\left(\sum _{j=0}^{\left\lceil \frac{a-1}{2}\right\rceil -1}z^{2cj}\right),which is a Kronecker polynomial.With the given {{x}}, we have the desirable {{s}}-di... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.017255960032343864,
0.037807490676641464,
-0.030072057619690895,
-0.023328348994255066,
0.014227394014596939,
0.005458284635096788,
0.050654102116823196,
-0.008536436595022678,
-0.042933929711580276,
-0.002172252396121621,
-0.04204900562763214,
0.00786511693149805,
-0.05492613464593887,
... | |
3112cb94cb384268a4a0e836dceaf21b083f951c | subsection | 43 | 78 | Four Main Theorems | We define a bijection \phi :\left\langle a-1 \right\rangle \times \left\langle a \right\rangle \rightarrow \left\langle a-1 \right\rangle \times \left\langle a \right\rangle by sending (i_1,i_2)\in A to the element (i_2,i_1+1)\in B and sending the element (i_1,i_2)\in B to the element (i_2-1,i_1)\in A.Now, using (REF )... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.028351908549666405,
0.02090533636510372,
-0.02668863721191883,
-0.012962833978235722,
0.009712588973343372,
-0.005020332522690296,
0.05163770541548729,
-0.009476068429648876,
-0.0036279151681810617,
0.024628622457385063,
-0.03613418713212013,
0.0019722734577953815,
-0.031586892902851105,
... | |
a5e5362f66a85b078c024a123f3cdda8da4372bf | subsection | 44 | 78 | Four Main Theorems | For convenience, for any {{i}}\in \mathbb {Z}^2, we letu({{i}}) := c i_1 + (ac-1) i_2 - \left\lfloor \frac{i_1-i_2}{a} \right\rfloor .It is straightforward to verify that for any m=1,2,\dots , a-1, we haveu( ma-1, a-1) = u (a^2-1, m-1).Notice thatI^{\prime } := \left\langle a^2 \right\rangle \times \left\langle a \righ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.020616192370653152,
0.013001477345824242,
-0.0222795270383358,
-0.0024854643270373344,
0.026918550953269005,
0.016679124906659126,
0.022722065448760986,
0.015290469862520695,
-0.006508368533104658,
0.022126926109194756,
-0.0509987510740757,
-0.005974270403385162,
-0.018083039671182632,
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.