01 / UNDERSTAND THE CONSTRUCTION

From interactions to choices.

Start with four items. Change who affects whom, compress the matrix, and watch the choice probabilities move.

Live calculation · no dataset needed

Selected rank2coordinates per vector

Matrix reconstruction RMSE—12 used cells only

Raw factor coordinates162 × 4 × rank

STEP 1 · SET THE EFFECTS

Interaction matrix U

A row is the target; a column is the context. U_AB = how B changes A's utility. Positive pushes A up; negative pulls A down.

The diagonal is unused: an item does not act as context for itself.

Examples

STEP 2 · COMPRESS

Choose a rank

2

For target x and context zÛ_xz = t_x · c_zOne dot product per directed effect

STEP 3 · INSPECT THE FACTORS

Two roles for every item

Û = T C^⊤

Each item has a target vector (how it responds) and a context vector (how it influences others). Their coordinates can change while the reconstructed effects stay the same.

Target vectorsT · 4 × 2

Context vectorsC · 4 × 2

Reconstructed interactionsÛ · 4 × 4

COMPARE 1–4

What gets lost?

off-diagonal RMSE ↓

Rank 3 can match all twelve used cells here because the four diagonal entries are free. A zero matrix error says nothing yet about predicting new choices.

STEP 4 · FORM A CHOICE SET

Who gets chosen?

A and B stay in the set. Add C or D and see each item's utility and softmax probability.

AB C D

ORIGINAL MATRIX

RANK 2 APPROXIMATION

Matrix fitting above uses least squares on the 12 known entries. The experiment lab below fits choice likelihood from observed winners, as in a statistical choice model.

02 / RUN A SEEDED SYNTHETIC STUDY

Can the model learn from choices?

Generate observed winners, fit MNL and CDM models, and select rank using validation NLL.

Browser computation · reproducible seed

DATA GENERATION

Experiment settings

Choice data source

ItemsChoices

Choice sets

Effect scaleRestarts per rank

Random seed

Try

Ready. The same generated data and split will be used for every fitted model.

TRAIN · VALIDATE · TEST

Results

Your experiment starts here

Choose a data source and run it. The chart will show how held-out validation NLL changes with rank.

WHAT THE NUMBERS MEAN

How to read the run

Train

Fit parameters using 70% of the observed choices.

Validation

Compare ranks on a separate 15%. Lowest NLL selects the rank.

Test

After selection, evaluate once on the final 15%.

NLL = average −log(probability given to the chosen item). Lower is better. Each rank has new random initial factors for each restart; the data stay fixed. Fits use Adam with a small L2 penalty and early stopping. The scaling study instead fits full CDM at four nested sample sizes, repeats three trials, and plots mean centered matrix MSE with a 1/m guide.

03 / THE PAPER CONNECTION

From the paper to a worked example.

Read the CDM construction, then follow one small dataset through ranks 1, 2, and 3.

About this guide. The Chai–Coffee–Juice worked example and its explanation are adapted from the original study notes of Nitesh Kumar Shah, developed under the supervision of Prof. Suryanarayana Sankagiri at IIT Madras. The CDM and low-rank model come from Seshadri, Peysakhovich, and Ugander (ICML 2019). These toy observations and initial factors are teaching examples, not data or results from the paper.

01 / CDM

Directed interactions

The paper writes u_xz for context item z's contribution to target item x. Add contributions from the other items in the offered set:

utility(x | S) = ∑_z∈S\{x} u_xz

Then softmax turns all item utilities into probabilities. Editing the matrix in the first view changes this calculation immediately.

02 / LOW RANK

Learn two vectors per item

The low-rank construction replaces a separate number for every directed pair with a target vector and a context vector:

u_xz = t_x · c_z

Rank r means r coordinates in each vector. The matrix view fits those vectors to an editable matrix; the experiment view learns them from simulated choices.

03 / IIA

What MNL cannot change

In MNL, adding C to {A, B} cannot change the odds P(A)/P(B). CDM can change those odds because C may affect A and B differently.

IIA: P(A | {A,B}) / P(B | {A,B}) = P(A | {A,B,C}) / P(B | {A,B,C})

The matrix studio shows these odds from the current interactions.

04 / HELD-OUT FIT

Why compare on new choices?

The paper compares out-of-sample negative log-likelihood for MNL and CDMs of different ranks on real data. Here you can recreate the question with new, seeded synthetic choices.

NLL = −mean log P(chosen item | offered set)

A lower training NLL alone does not establish that a higher rank predicts better. Inspect validation, then the final test comparison.

FROM NITESH'S ORIGINAL NOTES · WORKED EXAMPLE

Chai, Coffee, and Juice

u_xz = t_x · c_z

The same six observed choices stay fixed. Select an illustrative rank and one observation to see how target and context vectors produce interactions, scores, and the probability of the chosen item. The factor values are starting examples; they were not learned from these six choices.

Choose rank

Choose observed choice

01 / INITIAL FACTORS

Two roles per item

Each row has a target vector t and a context vector c. More coordinates mean more products in each interaction.

02 / FACTOR PRODUCT

Directed interactions

Rows are targets; columns are contexts. The diagonal is shown for the product, but is never used as an interaction in a choice.

03 / ONE OBSERVATION

PROBABILITY OF OBSERVED CHOICE

04 / SAME SIX CHOICES

Compare the initial values across ranks

Average NLL = −(1/6) ∑ log P(chosen item | offered set). Lower describes these particular values; it does not establish which rank would fit best after training. For only three items, rank 3 illustrates the arithmetic rather than parameter compression.

SCOPE OF THIS APP

Inspired by the paper, not a numerical reproduction

The experiment lab generates fresh synthetic data with adjustable settings. Its curves are not the paper's Figure 1–3, and it does not load SFWork, SFShop, nature photos, or the paper's mixed MNL comparison. Numerical results will differ with data, seed, sample size, optimization, and regularization.

For a faithful reproduction, use the original datasets and exact preprocessing, splits, baselines, optimization settings, and repeated trials. The app is a place to understand the mechanics before doing that work in a research notebook.

Open the ICML 2019 paper ↗