One code. Every size. Every group.

Many AI systems compress data into a short code and add more numbers when a bigger budget is available, so the first few numbers have to work on their own. This is called a nested representation; Matryoshka embeddings work this way. My paper asks what that convenience costs when one code has to serve several populations, and we judge it by the group it serves worst.

Best at each size is not the same as best for all sizes.

If each size could choose its own directions, it would pick whatever protects the worst-off group at that size. A nested code cannot: the size-2 code has to keep the size-1 direction. The price of nestedness compares the nested code’s worst-group error with the best separate code, size by size.

Prove it first, then check it on real data.

The paper bounds how large the price of nestedness can get, builds examples that show when some price is unavoidable, and separates the cost of nesting itself from the cost of imperfect optimization. It then audits fitted data and pretrained nested text-embedding models with certificates: guaranteed ranges for the true cost, not just point estimates.

Working paper

The manuscript, its exact bounds, and its numerical results are not public yet. This page explains the question and a hand-worked toy example; it does not report the paper’s findings.

Smaller should not mean unfair.

Nested codes let one model serve a phone and a server, a quick search and a careful one. If shrinking the code quietly shortchanges one group, that cost should be measured and bounded, not assumed away. The same question runs through all of my research: how little can a system use and still keep its promise to everyone?

Not just what reviewers think, but how they write.

My first research paper, written the summer after ninth grade. Sentiment analysis sorts reviews into positive and negative. I asked whether writing style alone, with no labels, could reveal distinct kinds of reviewers, and whether those groups behave differently.

Turning style into numbers.

I measured 18 stylometric features from earlier personality research, such as punctuation, pronouns, long words, and vocabulary richness, then merged highly correlated features into 8 core ones. I clustered the reviews with K-Means, BIRCH, and Gaussian mixture models, chose five clusters with the elbow method, and named each cluster by reading samples from it.

Five ways to review a movie.

People mix styles, but patterns hold.

The personas overlapped heavily and clustering scores were modest, because reviewers rarely stick to one style. Patterns still held: emotional reviewers gave the most extreme ratings, personas leaned toward different genres (horror for the analytical groups, comedy and romance for emotional reviewers), and sentiment was balanced within every persona, so writing style is independent of whether a review is positive or negative.

Only K-Means separated Candid from Emotional reviewers; BIRCH and the mixture model merged them.

Where it led

This project started my research path. It looked at how different groups of people express themselves; my later work asks how AI systems can serve every one of those groups, with guarantees.

Learning orders. Bounds decide.

ExactPHQ asks a narrow question: can fewer PHQ-9 items determine the same binary threshold class as all nine? A learned policy chooses what to ask next. A deterministic certificate alone decides when to stop.

These are retrospective question counts and exact fixed-response score classification, not clinical equivalence.

The stopping certificate

For answered items, L is the observed sum. With m unanswered items scored 0–3, U = L + 3m. At T = 10, stop positive if L ≥ T; stop negative if U < T. Otherwise another response is necessary. Model confidence cannot narrow these bounds.

Item 9 is captured first. That protocol does not establish suicide-risk assessment or a safety intervention.

Freeze, replay, replicate.

We developed policies on adult NHANES 2017–2018 records, locked them, and replayed complete responses on two different cycles. The newer cycle is the primary external evaluation; the earlier cycle adds replication.

Empirical dynamic programming plans future question cost. Secondary policies use a categorical MADE density or a compact latent-class mixture. Learning changes the order, never the stopping rule. Later searches are exploratory because they reuse already examined cohorts.

A smaller learning gain, honestly shown.

Most savings come from certified stopping and a good fixed order. In the primary cohort, empirical DP saves only 0.010 questions beyond optimal fixed ordering. Exploratory MADE saves 0.090 beyond fixed and 0.041 beyond an independent-item model.

All 4⁹ fixed patterns had zero class errors in independent exhaustive traces. Preserving five severity bands instead required 8.394 questions in the newer cohort.

What the guarantee covers

The guarantee is exact binary score emulation for a fixed response vector. It does not say a person will give the same responses after reordering. A prospective study would need to measure response consistency, time, dropout and distress, with independent human escalation rules.

• No claim of diagnosis, time savings, or prospective clinical equivalence.
• The binary threshold guarantee does not preserve an exact total or complete severity assessment.
• The work was coauthored by Brandon Baek, Tevyn Yong and Seonho Kim.

A change can disappear before the next check.

RECHECK-ED studies when to place one already permitted flexible nursing reassessment. It combines recent waveform support with the probability that a visible change ends within 20 minutes, while keeping required checks fixed.

A retrospective information-capture study, not a trial of outcomes, nurse workload, or safety.

Timing without moving the protected checks

Two patient-separated, five-member gradient-boosted ensembles estimate waveform-rule support q and 20-minute disappearance e from past information only. Their product q × e forms a short-horizon urgency heuristic. Missing waveforms cannot reassure the controller: an active raw-monitor event triggers a check.

Patients separated. Method frozen.

The method was developed on 450 previously opened MC-MED waveform episodes across 412 patients. Model files, thresholds, code and waveform inventory were frozen before a one-time evaluation on 600 new episodes from 567 different patients.

The other 202 waveform-qualified episodes were already chart-represented. Thirty-one of the 600 episodes had no complete waveform overlap. The case-enriched, single-center sample does not estimate general ED event prevalence.

Capture and selectivity are different gains.

The raw-monitor trigger captured the same 70 episodes as the model. Learning’s incremental result was 56 endpoint-negative triggered actions instead of 67—not fewer total attempted checks. Retrained no-waveform and process-only models also captured 70, with 62 and 60 endpoint-negative prompts.

A research prototype, governed by nurses.

The result supports monitor-responsive timing and a smaller set of endpoint-negative prompts under a fixed experimental budget. It does not establish that nurses missed events, that prompts were unnecessary, or that waiting would be safe. Required, urgent, post-medication and post-procedure checks cannot be moved.

• PPG pulse quality cannot confirm arterial oxygen saturation.
• The replay leaves physiology, treatment and historical documentation unchanged.
• Prospective work needs nursing review, local permissions, override rules and measured patient outcomes.